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Doctoral Thesis

Multivariate extremes and regular variation for stochasticprocesses

Author(s): Lindskog, Filip

Publication Date: 2004

Permanent Link: https://doi.org/10.3929/ethz-a-004669275

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Diss. ETH No. 15319

Multivariate Extremes and

Regular Variation for

Stochastic Processes

A dissertation submitted to the

SWISS FEDERAL INSTITUTE OF TECHNOLOGY

ZURICH

for the degree of

Doctor of Mathematics

presented by

FILIP LINDSKOG

MSc in Engineering Physics KTH

born 20.10.1975

citizen of Sweden

accepted on the recommendation of

Prof. Dr. P. Embrechts, examiner

Prof. Dr. F. Delbaen, co-examiner

2004

to my parents

Acknowledgement

First I want to thank my supervisor Paul Embrechts for having confi¬

dence in me and for his constant encouragement. I am most grateful

to him for giving me the time and freedom needed to pursue the some¬

times vague ideas I believed would lead to interesting results. I also

want to thank my friend and colleague Henrik Huit for our cooperation

during the past years. I have benefited greatly from our joint work and

discussions during Henrik's visits to ETH Zürich and my visits to the

Mathematical statistics division at KTH in Stockholm. A special thank

goes to Alexander McNeil and Uwe Schmock for fruitful discussions and

joint work in various forms during the past years. Furthermore, I want

to thank Freddy Delbaen for being co-examiner and for his many inter¬

esting comments on the thesis. I am also very grateful to Credit Suisse,

Swiss Re and UBS for financial support through RiskLab (Switzerland)and for giving me the possibility to combine my work at RiskLab with

my PhD studies. Without this possibility this thesis could not have

been written. Finally I want to thank my colleagues at ETH Zürich for

their friendship and support.

v

Contents

Abstract ix

Kurzfassung xi

Introduction 1

1 Multivariate regular variation 9

1.1 Vague convergence 10

1.2 Vague convergence on the state space R \{0} 12

1.3 Multivariate regular variation 18

1.4 Sums of regularly varying random vectors 28

2 Regular variation for multivariate additive processes 35

2.1 Additive processes 38

2.2 Regular variation for multivariate additive processes and

for vectors of functionals acting on such processes ....41

3 Regular variation for general stochastic processes 65

vii

viii Contents

3.1 Regular variation on D 69

3.1.1 Proofs 73

3.2 Markov processes with asymptotically independent incre¬

ments 78

3.2.1 Proofs 85

3.3 Filtered Levy processes 91

4 Dependence in elliptical distributions 103

4.1 Elliptical distributions 104

4.2 Kendall's tau and Spearman's rho for elliptical distributions 109

4.3 Proof of Theorem 4.14 112

4.4 Proof of the counterexample 116

5 Multivariate extremes for elliptical distributions 125

5.1 The connection between regular variation and tail depen¬

dence 126

5.2 Interpretations and explicit computations of spectral mea¬

sures 131

Bibliography 138

Curriculum Vitae 143

Abstract

In this thesis extremes for random vectors and multivariate stochastic

processes are studied using the notion of multivariate regular variation

and its extensions to state spaces suitable for heavy tail analysis for

continuous-time stochastic processes. For multivariate stochastic pro¬

cesses X = (Xf)t>o questions concerning the implications of regular

variation of Xj for some fixed t > 0 on the tail behavior of various vec¬

tors of functionals acting on X are addressed. Formulations of regular

variation for the graph of a stochastic process and of regular variation

on the space of right-continuous functions with left limits are studied

and used to characterize the extremal behavior of stochastic processes

and to determine the tail behavior of vectors of functionals acting on

stochastic processes satisfying some of the regular variation conditions

considered. The formulation of regular variation on the space of right-

continuous functions with left limits is found to be particularly useful,

providing a powerful and elegant approach to heavy tail analysis for

stochastic processes similar to the classical weak convergence approach

of Billingsley [5].Another topic of this thesis is the study of various notions of dependence

and multivariate extremes for elliptical distributions. This includes the

concordance measures Kendall's tau and Spearman's rho, tail depen¬

dence coefficients and spectral measures with respect to different norms,

for which explicit expressions are computed.

ix

Kurzfassung

In dieser Dissertation werden Extreme von Zufallsvektoren und multi-

variaten stochastischen Prozessen untersucht. Hierbei benutzt wird das

Konzept der multivariaten regulären Variation und dessen Erweiterun¬

gen auf Zustandsräume die passend sind für die Analyse des asymp¬

totischen Verhaltens von stochastischen Prozessen in stetiger Zeit. Für

multivariate stochastische Prozesse X = (X)f>o wird untersucht, wie die

reguläre Variation von Xt für ein festes t > 0 das asymptotische Verhal¬

ten von verschiedenen Vektoren von Funktionalen beeinflusst, die auf X

operieren. Es werden Formulierungen studiert der regulären Variation

für den Graphen eines stochastischen Prozesses, sowie für rechtsstetige

Funktionen mit linksseitigen Grenzwerten. Diese werden benutzt, um

das extreme Verhalten von stochastischen Prozessen zu charakterisieren,

sowie um das asymptotische Verhalten zu bestimmen von Vektoren

von Funktionalen, die auf stochastischen Prozessen operieren, welche

einer bestimmten Form von regulärer Variation genügen. Die For¬

mulierung der regulären Variation auf dem Raum der rechtsstetigen

Funktionen mit linksseitigen Grenzwerten scheint besonders nützlich.

Sie ermöglicht einen mächtigen und eleganten Zugang zur asymptotis¬

chen Analyse von stochastischen Prozessen, ähnlich dem klassischen Zu¬

gang der schwachen Konvergenz von Billingsley [5].Ein weiteres Thema dieser Dissertation ist das Studium von diversen

Begriffen von Abhängigkeit und multivariaten Extreme von elliptischen

Verteilungen. Dies beinhaltet Kendalls tau, Spearmans rho, Koeffizien¬

ten der "tail dependence" und Spektralmasse bezüglich verschiedener

Normen, für welche explizite Ausdrücke berechnet werden.

xi

Introduction

The tail behavior of heavy-tailed random variables and functionals of

univariate stochastic processes has been studied successfully for quite

some time using the notion of regular variation. The theory of regularly

varying functions is an important field of real analysis with links to

number theory and complex analysis, but also to probability theory. If

X is a random variable with distribution function F, then X is said

to be regularly varying (at oo) with index a > 0 if 1 — F is regularly

varying (at oo) with index —a, i.e. if for every x > 0, as u —> oo,

l^FM1 - F{u)

Here X is considered heavy-tailed due to the relatively slow power-

law-type decay of 1 — F(u) as « —)• oo. Regular variation of X has

interesting implications for the tail behavior of various functions of X

and for the tail behavior of sums and maxima of independent copies of

X. All essential results on univariate regular variation can be found in

Bingham, Goldie and Teugels [7] ; a well written extensive collection of

results written with a high level of mathematical rigour. Various formu¬

lations of multivariate regular variation have been studied and used to

obtain results for stochastic phenomena where heavy tails appear in a

wide range of applications, including multivariate extreme value theory,

the description of weak limits of point process, the study of solutions to

stochastic recurrence equations, financial risk management and many

more. Here one is interested in the limit measure as n —> oo of a se¬

quence of measures of the form nP(a~1X <G • ), where X is a random

1

2 Introduction

vector and an —>• oo. Roughly speaking, the random vector X is said to

be (multivariate) regularly varying if the above sequence has a nonzero

limit measure. There are many other possible formulations of multivari¬

ate regular variation, but most of them have in common that they are

expressed in terms of convergence of measures. There is at this point

no similar collection of results for multivariate regular variation. Re¬

sults on multivariate regular variation appear in various places in more

or less different mathematical settings and although most basic results

on multivariate regular variation come as no surprise to those familiar

with the classical univariate theory, there are essential differences and

important results which do not seem to appear in the literature. There¬

fore, in Chapter 1 we give a detailed presentation of important aspects

of and fundamental results for multivariate regular variation. We also

derive results on linear transformations and sums of (multivariate) reg¬

ularly varying random vectors; these results are used extensively in the

subsequent analysis.

In applications one sometimes encounters data sets or time series with

a few extremely large observations. For such data sets it might be

appropriate to use heavy-tailed probability distributions to model the

underlying uncertainty. This is the case for instance in so-called catas¬

trophe insurance (fire, wind-storm, flooding) where the occurrence of

large claims may lead to large fluctuations in the cash-flow process

faced by the insurance company. The situation is similar in finance

where extremely large losses sometimes occur, indicating heavy tails of

the return distributions. The probability of extreme stock price move¬

ments has to be accounted for when analyzing the risk of a portfolio.

Another application is telecommunications networks where long service

times may result in large variability in the workload process. In many

applications it is appropriate to use a stochastic process (Xt)t>0 to

model the evolution of the quantity of interest over time. The notion

of heavy tails enters naturally in this context either as an assumption

on the marginals Xt or as an assumption on the increments Xt+h — Xt

of the process. However, it is often the case that the marginals or the

increments of the process are not the main concern, but rather some

functional of the process. A natural example is the supremum of the

process over a time interval, supir0T-| Xt. Another example is the av-

Introduction 3

erage of the process over a time interval, T_1 JQ Xtdt. We are then

typically interested in the probability that the functional exceeds some

high level, e.g. -What is the probability that the sea level exceeds a high

barrier sometime during [0,T]? It may therefore be important to know

how the tail behavior of the marginals Xt (or the increments) is related

to the tail behavior of functionals of the process. A lot of effort has been

put into studying the tail asymptotics of the distribution of functionals

of heavy-tailed univariate stochastic processes. Interesting results are

found in Embrechts, Goldie and Veraverbeke [15] under the assumption

of subexponentiality. For univariate infinitely divisible processes results

on the tail behavior for subadditive functionals are derived in Rosihski

and Samorodnitsky [36] also under assumptions of subexponentiality.

See also Braverman, Mikosch and Samorodnitsky [9] for further results

on the tail behavior of subadditive functionals of univariate regularly

varying Levy processes. However, whereas problems concerning the tail

behavior of univariate heavy-tailed stochastic processes (or functionals

of such) have been studied successfully for quite some time, multivariate

processes have received far less attention. Even though the intuition be¬

hind the univariate results to a large extent extends to the multivariate

case, proving results in the multivariate case requires other tools. The

reason for this is that the analytic notion of regularly varying or subex-

ponential functions which is used in univariate heavy tail analysis turns

out to be quite awkward and difficult to use when extended to the mul¬

tivariate case. In short, multivariate distribution functions are for many

purposes not natural objects causing the multivariate extensions to be

far from elegant and transparent. An approach based on convergence

of measures (described in detail in Chapter 1) makes the difference be¬

tween the univariate and multivariate heavy tail analysis disappear. It

also lends itself to nice geometrical interpretations and reduces tedious

technicalities to a minimum. For further discussions about this issue we

refer to Resnick [35] and the references therein. In the multivariate case

one typically studies a d-dimensional stochastic process (X.t)t>o- The

process could be interpreted for instance as measurements of sea levels

at d different locations, high-frequency return data of d different stocks

or claims in d different insurance lines. Clearly, an important difference

between the multivariate case and the univariate case when analyzing

4 Introduction

extremes is the possibility to have dependence between the components

of the random vector Xf. Large values may for instance tend to occur si¬

multaneously in the different components. One example is an important

macroeconomic event causing simultaneous big price drops for several

stocks. To have a good understanding of the dependence between ex¬

treme events in the multivariate case may be of great importance in

applications. Similar to the univariate case some functional or vector

of functionals of the process may be the primary concern. Natural

examples are for instance the componentwise suprema of the process

(supfGr0Tn JQ ,..., supfGr0)Ti X\ ). Another example is the compo¬

nentwise average of the process. Other functionals or combinations of

functionals may also be of interest. We are typically interested in the

probability that the vector of functionals belongs to some set far away

from the origin, e.g. -What is the probability that the sea level exceeds

a high barrier at some (or all) locations sometime during [0,T]? To

answer this type of questions we need to know how the tail behavior of

the marginals Xf is related to the tail behavior of vectors of functionals

of the process. This is the topic of Chapters 2 and 3.

In Chapter 2 we address these questions when the underlying stochastic

process is a multivariate additive process, i.e. a stochastically continuous

process with independent increments starting at zero. More precisely,

we consider an additive process (X.t)t>o and we study the implications

of Xf being regularly varying for some fixed t > 0 on the tail behavior of

vectors of functionals depending on the sample path of the process up to

time t. For example, we find that regular variation of Xf implies regular

variation of (supiGr0T] XJ; ,... ,suptGr0T] JQ ) with a limit measure

that is fully determined by that of X^. For functionals more complicated

then e.g. the componentwise suprema, knowing only the tail behavior

of X^ is insufficient for determining the tail behavior of the functionals

of the sample path up to time t. Therefore, we study a formulation

of regular variation for the graph of a multivariate stochastic process

and we give necessary and sufficient conditions for an additive process

to have a regularly varying graph. We then use this formulation for

deriving sufficient conditions for regular variation of the random vector

(J0 Xs ds,..., J*0 Xs ds) when (Xs)sG[0,i] is a d-dimensional additive

process with a regularly varying graph, and we show how the respective

Introduction 5

regular variation limit measures are related.

In Chapter 3 we consider a different and more general approach to

heavy tail analysis for stochastic processes. Just as multivariate regu¬

lar variation provides a natural way for understanding the tail behav¬

ior of heavy-tailed random vectors, a similar formulation for stochastic

processes with sample paths in D([0,l],Md) - the space of Md-valued

right-continuous functions on [0,1] with left limits - provides us with a

natural and powerful framework for studying the extremal behavior of

heavy-tailed stochastic processes. The chapter can be divided into three

main parts. In the first part we give two formulations of regular varia¬

tion on .D([0,1], Md) and we show that they are equivalent. We then give

necessary and sufficient conditions for regular variation on D([0,1], Md)in terms of multivariate regular variation of the finite-dimensional dis¬

tributions and relative compactness. We also give a version of the Con¬

tinuous Mapping Theorem for this setting, a result which proves very

useful in the subsequent analysis. In the second part we consider strong

Markov processes (not necessarily time-homogeneous) with asymptoti¬

cally independent increments in a certain sense. We show that for such

processes the sufficient conditions for regular variation on D([0, l],Md)simplifies considerably; they are almost the same as the necessary and

sufficient conditions for having a regularly varying graph. Moreover, we

find that if a strong Markov process with asymptotically independent

increments and sample paths in D([0, l],Ed) is regularly varying, then

its regular variation limit measure concentrates on step functions with

one step. This means that the process hits a Borel set in Mr" far away

from the origin by making one big jump and, in comparison to the size

of the jump, does not move much before and after the jump. More¬

over, knowing that the regular variation limit measure concentrates on

step functions with one step we can explicitly compute the regular vari¬

ation limit measure of many interesting vectors of functionals acting

on the process by simply applying our version of the Continuous Map¬

ping Theorem. In the third part we consider processes Y which can

be expressed as an integral of a deterministic function / : [0, l]2 —> R

with respect to a regularly varying Levy process X. The idea here is

that we can compute the regular variation limit measure of h(X), where

h : D([0, l],Ed) -)- D([0, l],Rd) satisfies the conditions of the Continu-

6 Introduction

ous Mapping Theorem enabling us to derive the extremal behavior of

h(K) from that of X and the properties of /, and show that Y and

h(K) have the same regular variation limit measure. To exemplify this

idea, we explicitly compute the regular variation limit measure of an

Ornstein-Uhlenbeck process driven by a regularly varying Levy process.

We find that the framework set up and studied in this chapter is general

enough to apply to the majority of interesting problems that arise in

multivariate heavy tail analysis for stochastic processes, yet powerful

enough to produce explicit results.

The canonical example of a multivariate distribution with a nontrivial

dependence structure is the multivariate normal distribution. A large

number of multivariate models, in particular in mathematical finance,

are based on the multivariate normal distribution since it enables ef¬

ficient simulation and explicit computation of many interesting quan¬

tities. However, at a closer look one often finds that the multivariate

normal distribution and models based on it do not capture essential

properties indicated by the observed data. For example, the empiri¬

cal distribution of financial return data has typically much heavier tails

than the normal distribution. Moreover, there is a tendency of simul¬

taneous large negative returns indicating a strong dependence between

extreme returns which cannot by properly modelled by a multivariate

normal distribution. In Chapter 4 we study the most natural extension

of the multivariate normal distribution, namely the class of elliptical dis¬

tributions. The class of elliptical distributions provides a rich source of

multivariate distributions which share many of the tractable properties

of the multivariate normal distribution and enables modelling of joint

extremes and other forms of nonnormal dependences. We show how

one can use elliptical distributions to construct multivariate discrete

time stochastic processes having elliptical finite-dimensional distribu¬

tions. Moreover, we study the concordance measures Kendall's tau and

Spearman's rho which play an important role in parameter estimation

for models based on elliptical distributions. In Chapter 5 we analyze

multivariate extremes for elliptical distributions. We find that the gen¬

eral stochastic representation of elliptically distributed random vectors

enables us to explicitly compute interesting quantities such as coeffi¬

cients of tail dependence (which measure the strength of dependence of

Introduction 7

joint extremes) and spectral measures associated with regularly vary¬

ing random vectors (a spectral measure is a probability measure on the

unit sphere with respect to some norm; the probability assigned to a set

represents the likelihood of an extreme observation being located in the

corresponding directions). The analysis in this chapter also highlights

various aspects of the concept of multivariate regular variation intro¬

duced in Chapter 1. The explicit expressions for densities of spectral

measures derived offer possibilities to study the effect on the depen¬

dence of multivariate extremes when varying the correlation structure

and the tail index of the marginals. Moreover, we study the effect of

different choices of norms - corresponding to different criteria for what

constitutes an extreme multivariate observation - on how the spectral

measure assigns probability mass to different directions. For example,

X may be considered extreme either if |X|2 = \/XTX exceeds some

high threshold, or if |X|oo = maxj \X^\ exceeds some high threshold.

Comments on the thesis

The first three chapters of this thesis are based on the papers [23] and

[24] (with H. Huit). The last two chapters which are based on the papers

[28] (with A. McNeil and U. Schmock) and [22] (with H. Huit) are less

technical and represent parts of my work within RiskLab. Although

the thesis is based on the papers mentioned above many new results,

comments and examples have been added to the thesis. The material

presented in the first three chapters is structured and presented as it is

with the aim of presenting a general framework for heavy tail analysis

for stochastic processes as well as a collection of relevant and useful

results on this topic. Chapter 3 can be considered as the main chapter.

However, in order to fully appreciate the usefulness of the approach and

the results presented in Chapter 3 a comparison with the approach and

analysis of Chapter 2 is helpful.

Throughout this thesis we assume as given a probability space (Q, J7, P)on which all random elements are defined.

Chapter 1

Multivariate regular

variation

In this chapter we introduce the notion of multivariate regular variation

and give results for regularly varying random vectors and for sums of

such. The chapter is organized as follows. In Section 1.1 we review the

notion of vague convergence following Kallenberg [26] and Resnick [34].In Section 1.2 we focus on properties of vague convergence on state

spaces relevant for multivariate regular variation. In particular we give

sufficient conditions for vague convergence on these state spaces which

will be used in the subsequent chapters. In Section 1.3 we introduce the

notion of multivariate regular variation. In particular we give impor¬

tant properties of the limit measure associated with a regularly varying

random vector and we give an equivalent formulation of multivariate

regular variation which will facilitate interpretations of results derived

in the following chapters. In Section 1.4 we give results on the tail

behavior of sums of regularly varying random vectors. These results

are of independent interest but, more importantly, will play an impor¬

tant role in the following chapters when we study regular variation for

multivariate stochastic processes.

9

10 Chapter 1. Multivariate regular variation

1.1 Vague convergence

Let (.Ë, p) be a locally compact, complete and separable metric space

and let B(E) be the Borel a-algebra on E generated by the p-open sets.

For B C E we denote by B°,B and dB = B\B° its interior, closure

and boundary, respectively. If B C E, then (B, p) is a metric space and

B(B) = B(E)r\B,

where B(E)nB = {AnB :Ae B{E)} (see e.g. p. 224 in Billingsley [5]).A set B C E is said to be bounded or relatively compact if its closure

B is compact. Let C^-(E) denote the class of all continuous functions

/ : E —)• K_|_ = [0,oo) with compact support, i.e. if / G C^-(E), then

there exists a compact set K such that f(x) =0 for x G Kc. Let M+(E)be the class of all Radon measures on (E, B(E)), i.e. of all (nonnegative)measures fi such that fi(B) < oo for all relatively compact B G B(E).In order to discuss convergence of Radon measures on (E,B(E)) we

must topologize M+(E). The class of all finite intersections of M+(E)-sets of the form {/i G M+(E) : fE f(x)p(dx) G (s,t)} with arbitrary

/ G Ck(E) and s,t G M form a base for a topology on M+(E) which

is called the vague topology. The space M+(E) with this topology can

be metrized as a complete and separable metric space, see e.g. Kallen¬

berg [26] for an example of a possible metric. We note that a sequence

(/in), fin G M+(E), converges to fi G M+(E) in the vague topology,

written /j,n A- //, if and only if JE f{x)fin{dx) —> JE f(x)fi(dx) for ev¬

ery / G Ck(E). When considering the subspace consisting of all finite

measures on M+(E), then we obtain the so-called weak topology by

replacing C^-(E) by the class of all bounded and continuous functions

f : E —> K_|_. Similarly, (fin) converges to /i in this topology, written

pn A- /i, if and only if JE f{x)fin{dx) —> JE f(x)/i(dx) for every nonneg¬

ative, bounded and continuous /. A theoretical justification for looking

at sets rather than integrals is given by the following result.

Theorem 1.1 (Kallenberg [26], 15.7.2) Let p, /ii, fi2, • • •be Radon mea¬

sures on (E,B(E)). Then the following statements are equivalent.

(l) fln A fl,

1.1. Vague convergence 11

(ii) pn(B) —> p(B) for every relatively compact B G B(E) such that

p{dB) = 0.

(iii) limsupn^00 pn(F) < p(F) and liminfj^oo pn(G) > p(G) for ev¬

ery compact F G B(E) and every open relatively compact G G

B(E).

Although studying integrals rather than sets may lead to a sometimes

more elegant analytic approach, we prefer the latter since this leads to

more intuitive geometric interpretations.

Remark 1.2 Note that \imsup pn(F) < p(F) for every compact F if

and only if liminf pn(G) > p(G) for every open relatively compact G.

This can be shown as follows. Suppose that \imsup pn(F) < p,(F) for

every compact F. Take an arbitrary open relatively compact G. Then,

for some e > 0, D = {x G E : p(x, G) < e} is compact, p(dD) = 0 and

p(G) = p(D) - p(D\G) < \im pn(D) - limsup pn{D\G)n

n

< nmpn{D) - liminf pn{D\G) = liminf(/*„(£>) - pn{D\G))n n n

= liminf pn{G).n

Conversely, suppose that liminf pn{G) > p(G) for every open relatively

compact G. Take an arbitrary compact F. Then, for some e > 0,

D = {x G E : p(x, F) < e} is open and relatively compact, p(dD) = 0

and

p{F) = p(D)-p(D\F)>\impn{D)-\immîpn{D\F)n n

> \im pn(D) - lim sup pn(D\F) = limsup(/in(D) - ^n(D\F))n

n n

= \rmsup pn(F).n

Remark 1.3 If p,pi,p2, • • • are finite measures on (E,B(E)), then

pn A- p if and only if pn A- p and pn(E) —> p(E) (Kallenberg [26],

15.7.6).

We will frequently show vague convergence for a sequence (pn), P-n £

M+(E), by showing that the set {pn} is relatively compact in the vague

topology on M+(E) and that any two subsequential vague limits coin¬

cide. Therefore, we will need the following result.

Theorem 1.4 (Kallenberg [26], 15.7.5) A subset II of M+(E) is

relatively compact in the vague topology if and only if

sup p{B) < oo

neu

for every relatively compact B G B(E).

1.2 Vague convergence on the state space

Wd\{o}

Let | • | be any norm on Md and let Vß denote the Borel u-algebra gen¬

erated by the open sets with respect to the metric |x — y|, i.e. 7ld

is the usual Borel cr-algebra on Md. For x G Md and e > 0, let

B*,e = {y £ ^d ' |x — y| < e}. For x G R, let \x\ denote the ab¬

solute value of x. We will study vague convergence of measures of the

form n P(a~1X G • ) as n —> oo, where X is an Revalued random vector

and 0 < an t oo. Clearly, for every e > 0, B0e is relatively compact in

Md but nP(a~1X G -B0,e) ~* °°- Moreover, for an appropriate choice of

(an) we would expect (nP(a~1X G Bq J) to converge, but since vaguely

convergent measures only ensure convergence for relatively compact sets

this can not be established with this topology. To solve this problem we

will modify the state space Md and equip the modified state space with a

topology which renders sets B G 7ld bounded away from 0 (0 ^ B) rel¬

atively compact. Multivariate regular variation is typically formulated

in terms of vague convergence of Radon measures on M \{0}, where

M = [oo, oo]. We will show that M \{0} can be equipped with a metric

which renders M \{0} a locally compact, complete and separable metric

space. Moreover, with this choice of metric

nd n (Md\{o}) = £(Rd\{o}) n (Md\{o}),

1.2. Vague convergence on the state space M \{0} 13

j Cl

i.e. on Ed\{0} the Borel cr-algebra B(M \{0}) coincides with the usual

Borel cr-algebra 7ld, and every B G lZd bounded away from 0 is relatively

compact. Note that the measures we will consider assign zero mass to

M \Md ; the points of M \Md are of no interest apart from being a part

of the modification of the state space which enables us to use the notion

of vague convergence.

Theorem 1.5 There exists a metric p such that

(i) (M \{0},/)) is a locally compact, complete and separable metric

space and

(ii) if B(M \{0}) denotes the Borel a-algebra generated by the ~p-open

sets, then Ud n (Rd\{0}) = #(Rd\{0}) n (Rd\{0}).

Proof. Let W\{0} = (0, oo] xS^1, where S^1 = {x G Md : [x^ = 1}denotes the unit sphere with respect to the max-norm | • |oo given by

|x|oo = max^a^1)],..., |#(d)|). For x G Rd\{0} we write x = (x*,x).

Equip Rd\{0} with the metric

p(x,y) = max(|l/z* - l/y*\, |x - y|).

Since (0, oo] is complete and separable with the metric \l/x* — 1/y*| and

since S^"1 is complete and separable with the metric |x — y| it follows

that Rd\{0} is complete and separable with the metric p. Moreover,

Rd\{0} is locally compact; for every x G Rd\{0} there exists an e > 0

such that Bx e= {y G Rd\{0} : p(x, y) < e} is compact. Let

T: (Rd\{0},ftdn(Rd\{0}))

- ((0, oo) x 8d^\B(W\{0}) n ((0, oo) x S^1))

be given by T(x) = (Ix^x/lx^). Then T is continuous, one-to-one

and onto and T~l given by T_1((£*,x)) = x*5i is continuous. Hence,

there is a one-to-one correspondence between the subspace topologies on

Rd\{0} and (0, oo) x S^"1. With the convention arctan(ioo) = ±tt/2

and tan(±7r/2) = ±oo, let / : Rd\{0} - W\{0} be given by

/(x) = (|x|00,5r(x)/|p(x)|00),


where p(x) = (-| arctan^1)),..., ^ arctan(a;(d))). Then / is one-to-one

and onto and

/-1((£*, x)) = (tan(arctan(x*)x(-1-)),... ,tan(arctan(x*)x(-d-))).

Equip R \{0} with the metric /j(x,y) = p(/(x),/(y)). It is easily

shown that R \{0} is separable and locally compact. We now show

that it is complete. Let (xn) be a Cauchy sequence in R \{0}, i.e. (xn)satisfies p(f(xn), /(xm)) —y 0 as m, n —)• oo. Then (/(xn)) is a Cauchy

sequence in Rd\{0} and by completeness there exists z G Rd\{0} such

that p(/(xn),z) —)• 0 as n —> oo. Since / is onto there exists x such

that z = /(x) and hence p(f(xn), /(x)) —>• 0 as n —> oo. Hence, R \{0}is complete. By a similar argument one can show that / and /_1 are

continuous. Moreover, the restriction

/:(Rd\{0},^(Rd\{0})n(Rd\{0}))- ((0, oo) x s^1, B(W\{o}) n ((0, oo) x s^1))

of/ is continuous, one-to-one, onto and has a continuous inverse. Hence,

also T~lof is continuous, one-to-one, onto and has a continuous inverse.

It follows that

nd n (Rd\{o}) = £(Rd\{o}) n (Rd\{o}),

i.e. the Borel sets we are interested in are the usual ones. D

We call a subclass U oïB(E) a convergence-determining class if pn{B) —>•

p{B) for every /i-continuity set B G U implies pn A- p on B(E). We

now present some useful examples of convergence determining classes

for vague convergence on B(M \{0}).

d

Theorem 1.6 Let p, pi, P2, • • •be Radon measures on B(M \{0}) with

yu(Rd\Rd) = 0 and letU be a subclass ofUd n (Rd\{0}) such that

(i) U is closed under the formation of finite intersections and

(ii) each open set in Md bounded away from 0 is a finite or countable

union of elements ofU.

J

1.2. Vague convergence on the state space R \{0} 15

If pn(A) ->> p(A) for every AeU, then pn A p on B(M \{0}).

Proof. If Ai, ..., Am lie in U, then so do their intersections; hence, by

the inclusion-exclusion formula,

fj,n(UZiAi) = J2^(Ai)-J2^AinAj">i i<j

+ J2 fJ-niAiHAjHAk)-...i<j<k

-+ Y^KAiï-^KAinAj)i i<j

+ J2 KAinAjHAk)-...i<j<k

= K^ÏLiAi).

If G is open and relatively compact in R \{0}, then GnRd is open and

relatively compact in R \{0} and open and bounded away from 0 in

Md. Hence, GflRd = UiAi for some sequence (Ai) of elements of U and

p(G) = p(G n Md). Hence, for any e > 0, we can choose m such that

p(Ui<mAi) > p(G) - e. Thus,

liminf/iTl(Gf) > liminf pn(\Ji<mAi)n—^-oo n—>-oo

= p(yJi<mAi) > p(G) - e.

Since e was arbitrary, liminf^^oo pn(G) > p(G). The conclusion follows

from Theorem 1.1 and Remark 1.2. D

Remark 1.7 Note that Theorem 1.6 still holds if R and Md are re¬

placed by R+ and R^_.

We now present convergence determining classes that will prove useful

in the subsequent analysis. Typically we will prove convergence to a

limit measure p which we know has the scaling property of Theorem

1.8 below. The convergence determining classes presented below remain

convergence determining classes without the additional condition on the

limit measure p, but to avoid unnecessary lengthy arguments we prove

the results below under this stronger assumption.

Let | • | be any norm on Md and let §d_1 = {x G Md : |x| = 1}. For

u > 0 and S G B^'1) let Vu,s = {x G Md : |x| > w,x/|x| G S}.

Theorem 1.8 If p is a Radon measure onB(M \{0}) with p(M \Md) =

0 such that for some a > 0 p(uB) = u~ap(B) for every u > 0 and

5 G #(Rd\{0}); tfien

(%) /i(wSd_1) = 0 for every u > 0,

(ii) /i({x}) = 0 for every x G R \{0};

(Hi) p(dVUjS) = p(Vu,ds) for every u > 0 and S G #(§d_1);

(^y) /^(Vo^s}) = 0 for all but at most countably many s G Sd~1.

Proof, (i) Suppose that there exist u,c > 0 such that p(uSd~1) = c.

Then, for v > u,

/i(Vu>s--i\V0>s«.-i) > /iHeQnc«,«]^-1) = ^ M«®*-1)çGQn(u,t;]

> c(v/u)-a J2 X = °°-

Since Fu gd-^V^ §d-i is relatively compact this is a contradiction and

we conclude that /z(n§d_1) = 0.

(ii)By(i),M({x})</i(|x|§d-1) = 0.

(iii) By (i) and since /i(Rd\Rd) = 0, p(dVu,s) = p(uS) + p(Vu,ds) =

MK,ös)-(iv) Since p is a Radon measure p(Vißd-i) < oo and hence p(Vi^sy) = 0

for all but at most countably many s G Sd~1. By the scaling property,

p(Vi^sy) = 0 if and only if p(Vu^sy) = 0 for every u > 0. Letting u 4- 0,

the conclusion follows. D

Theorem 1.9 Let p, pi, P2-, • • •be Radon measures on B(M \{0}) such

that p satisfies the conditions of Theorem 1.8. If V = {VUjs ' u > 0, S G

1.2. Vague convergence on the state space R \{0} 17

B(Sd-1),p(dVUjS) = 0} and if pn(A) -> p(A) for every A e V, then

pnA p onB(Md\{0}).

Proof. Let U0 = {VUtS\VVtS 0 < u < v,S G B^'1)}. Clearly

Uq satisfies the conditions of Theorem 1.6. By Theorem 1.8 also U =

{A G Uo : p(dA) = 0} satisfies the conditions of Theorem 1.6 (U con¬

tains those elements of Uo that remain after removing the elements

Vu,s\Vv,s f°r which VUjds has a nonempty intersection with one of the

at most countably many rays Vo,{5} charged by p). By Theorem 1.8,

Vu,s\Vv,s G U if and only if K,s, Vv,s G V and if //„(K,s) - MK,s)and /^(K,s) - m(K,s), then

Mn(K,s\K,s) = Vn(Vu,s) ~ Vn(Vv,s)

-+ KVu,s) - KVv,s) = v(Vu,s\Vv,s)-

Hence, if pn(A) —* p(A) for every A G V, then pn(A) —y p(A) for every

A eU which in turn implies that pn A p on B(M \{0}). D

For a, b G Md we write a < b if a^ < 6^^ for i = 1,..., d. For a < b we

write [a, b) = [a^\b^) x • • • x [a(d), b^). If /i satisfies the conditions

of Theorem 1.8 and if [a, b) is bounded away from 0, then by Theorem

1.8 (i) with | • | being the max-norm, p(d[a.,h)) = 0; i.e. all sets [a,b)bounded away from 0 are //-continuity sets. Clearly those sets also

satisfy the conditions of Theorem 1.6.

—dLemma 1.10 Let p: pi, P2, • • •

be Radon measures on B(M \{0}) such

that p satisfies the conditions of Theorem 1.8. Let U be the collection

of sets of the form [a, b) such that a, b G Rd\{0}7 a p(A) for every A EU, then pn A p

onB(Md\{0}).

Clearly one can replace the collection of such half-open intervals by some

arbitrary combination of open, closed and half-open intervals and the

resulting collection of sets will remain a convergence determining class

for vague convergence on B(M \{0}) to a limit measure of the type of

Theorem 1.8.

Note that since R+\{0} is equipped with the subspace topology, the

boundary of elements in U with nonempty intersection with the coor¬

dinate hyperplanes have empty intersection with the coordinate hyper-

planes (for R+\{0} equipped with the subspace topology of R \{0} we

have for example ([1,2] x [0,1])° = (1,2) x [0,1)). Hence U always

contains the sets [a, b) which have nonempty intersections with the co¬

ordinate hyperplanes. Hence, if we would replace each set [a, b) in U

with (a, b), then we would no longer have a convergence determining

class.

—dLemma 1.11 Let p, pi, P2, • • •

be Radon measures on B(M+\{0}) such

that p satisfies the conditions of Theorem 1.8. Let U be the collection

of sets of the form [a, b) such that a, b G R+\{0} and a < b. //

pn(A) -t p(A) for every AeU, then pn A p on #(R+\{0}).

1.3 Multivariate regular variation

The notion of multivariate regularly varying random vectors has ap¬

peared in several apparently different applications such as the study of

stochastic recurrence equations (see Kesten [27]), multivariate extreme

value theory (see e.g. Resnick [34]), the study of domains of attraction

of multivariate distributions (see Rvaceva [37]) and the description of

weak limits of point processes constructed from stationary sequences

of random vectors (see Davis and Hsing [13]). In recent years some

effort has been made to establish equivalence of the different notions

of multivariate regular variation (see Basrak [2] and Basrak, Davis and

Mikosch [4]) and as a result several equivalent definitions exist. We will

consider mainly two of them. The following definition is perhaps the

most useful one.

Definition 1.12 An Md-valued random vector X is said to be regularly

varying if there exist a sequence (an), 0 < an "f oo and a nonzero Radon

1.3. Multivariate regular variation 19

measure p on B(M \{0}) with p(M \Md) = 0 such that, as n —y oo,

nP(a"1XG •) Ap(-) onB(Md\{0}). (1.1)

Remark 1.13 A standard regular variation argument shows that if

(1.1) holds, then the sequence (an) is regularly varying with index 1/a,i.e. for every A > 0, a^xn]/an —* ^l^a as n —> oo.

If Definition 1.12 holds, then the limit measure p has the following

scaling property which will play an important role in the subsequent

analysis.

Theorem 1.14 // the conditions of Definition 1.12 hold, then there

exists an a > 0 such that p(uB) = u~ap(B) for every u > 0 and

BeB(Md\{0}).

Proof. For r > 0 and S G #(Sd_1), let K,s = {x G Rd : |x| >

r, x/|x| G S}. Fix S G jB(8d_1) such that for some r > 0 we have

p(dVrjs) = 0. Since p is a Radon measure such a set S G #(§d_1) exists.

Then p(duVr,s) = 0 for all but at most countably many u G [l,oo).Denote this set by U, i.e. U = {u £ [1, oo) : p(duVr,s) = 0}.

Suppose that p(Vr,s) > 0. Define / and g on (0,oo) by f(x) = P(X G

xVr,s) and g(x) = p(xVrjs)- Then, for every u G U, as n —> oo,

nf(uan) = nP(X G anuVr,s) -> n(uVr,s) = g{u).

For x > ai, let t = t(x) be the largest integer with a* < x. Since / is

nonincreasing, i.e. x < y implies f(x) > f(y),

f(uat+1)/f(at) < f(ux)/f(x) < f(uat)/f(at+ï).

However, the lower bound is (t/(t + l))(t + l)f(uat+i)/(tf(at)) which

tends to g(u)/g(l), for each u G U. Similarly for the upper bound.

Hence, for every u G U, as x —y oo,

f(ux)/f(x)^g(u)/g(l).

For arbitrary u > 0, let g*(u) = lim sup^.^^ f(ux)/f(x). Then, g*(u) =

g(u)/g(l) for u G U. Moreover, g*(u) < 1 for u > 1 since f(ux)/f(x) <

1 for u > 1 and x > 0. In particular, limsup^ g*(u) < 1. It now

follows by Theorem 1.4.3 (ii) p. 18 in Bingham, Goldie and Teugels [7]that there exists an a G R such that, for every u > 0, as x —y oo,

f(ux)/f(x)^u~a,

i.e. p(uVrjs) = u~ap(Vrjs) for every u > 0.

Suppose that //(V^s) = 0. We will show that this implies that p(Vu,s) =

0 for every u > 0. Suppose that there exists ro G (0,r) such that

^{Yr0,s) > 0. Then, by the above arguments, there exists r\ G (0,r)such that p(Vrijs) > 0 and p(dVrijs) = 0. However, we must then have

p(uVrijs) = u~ap(Vri^s) for every u > 0. In particular, 0 = p(Vr,s) =

(r/ri)~ap(Vri,s) > 0 which is a contradiction.

Hence, for each yit-continuity set Vrjs there exists an a G R such that,

for every u > 0, //(uV^s) = w_a/i("\/rj£').It remains to show that a does not depend on Vr,s- This can be shown

by the same arguments as in Resnick [34] p. 277. Since the /i-continuity

sets of the form Vr,s for r > 0 and S G B(E>d~l) form a 7r-system

which generates B(M \{0}) fl R the scaling property holds for every

B G #(Rd\{0})nRd. However, since //(Rd\Rd) = 0 the scaling property

holds for every B G B(M \{0}). Since p is a Radon measure we must

have a > 0 and since p(M \Md) = 0 we must have a > 0. D

The following equivalent formulation of multivariate regular variation

will provide many nice interpretations of various regular variation re¬

sults in the following chapters.

Theorem 1.15 Let X be an Md-valued random vector. Then the fol¬

lowing statements are equivalent.

(i) X is regularly varying in the sense of Definition 1.12.

(ii) There exist an a > 0 and a probability measure a on #(8d_1) such

that, for every x > 0, as u —y oo,


Remark 1.16 If (i) holds, then by Theorem 1.14 there exists an a > 0

such that p(xB) = x~ap(B) for every x > 0 and B G #(Rd\{0}) and

(ii) holds with the same a. If (ii) holds, then (i) holds and p satisfies

the scaling property above with the same a (see the proof of Theorem

1.15 for details).

Definition 1.17 For a random vector X satisfying (1.2) we refer to a

and a as the tail index of X and the spectral measure of X with respect

to the norm \ \, respectively.

An immediate consequence of Theorem 1.15 is the following equivalent

formulation of regular variation for a nonnegative random variable.

Corollary 1.18 A nonnegative random variable R is regularly varying

(at oo,) with index a > 0 if and only if for every x > 0,

..

M(R > ux)hm J.

„f- = x .

u^oo F(R > u)

Example 1.19 Let X = RTJ, where the nonnegative random variable

R and the §d_1-valued random vector U are independent. Suppose that

R is regularly varying with index a > 0. Then

P(|X|>^,X/|X|G-) ¥(R > ux)

P(|X| >u) ¥(R>u){ )

^y x-aF(Ue-)

on #(§d_1) as u —y oo, i.e. X is regularly varying with index a and

spectral measure P(U G • ) with respect to the norm | • |.

For more interesting (nontrivial) examples, see Chapter 5.

Proof of Theorem 1.15. (ii) => (i) For x > 0 and S G B^'1), let

VXtS = {xGRd :|x| >x,x/|x|eS},

V = {V^s'-xyO^SeB^-1)}.

For u, x > 0 define the measures

Hu(Vx,) = P(|X|>«x,X/|X|G.)/P(|X|>«),

p(Vx,) = x'aa(-)

on #(§d_1). This also defines, uniquely, set functions pu and p on the

semiring V. By Theorem 11.3 p. 166 in Billingsley [6], pu and p can be

uniquely extended to measures on a(V) = IZd D (Rd\{0}). By requiring

that pu(M \Md) = p(M \Md) = 0, pu and p can be uniquely extended

to (Radon) measures on B(M \{0}). By definition of p, p(VXis) =

x~ap(Vi:s) for every x > 0 and S G #(§d_1). Suppose that there exist

x, c > 0 such that p.(dVx§d-i) = c. Then, for y > x,

M^s^-AK/,^-1) > KuqeQn(x,y]dVqjSd-i) = 22 M^V^d-i)qe<Qn(x,y]

> c(y/x)~a J2 1 = °°-

qeQn(x,y]

Since ^gd-i^gd-i is bounded in R \{0} this is a contradiction and

we conclude that p(dVx §d-i) = 0 for every x > 0. This implies in partic¬

ular that p(dVXjs) = p(Vx,ds) for every x > 0 and 5 G #(§d_1). Since

MuO^,-) -^ Kvx,-) for every a; > 0 this implies pu(VXjS) -> ß(Vx,s) as

u —y oo for every /i-continuity set VXjs- Since pu(M \Md) = p(M \Md) =

0, the //-continuity sets of V form a convergence determining class,

i.e. pu A- p on #(R \{0}). Let F denote the distribution function of |X|

and, for n > 1, let an = F"1^ - 1/n) = inf{s G R+ : F(s) > 1 - 1/n}.

Then, as n —y oo, nP(|X| > an) —>• 1 and

"F'!;'x''' = *.. (•) a mo on ß(sd\{o}),nP(|X| > an)

i.e. nP(a"1X G • ) -^ ju(-) on #(Rd\{0}).(i) => (ii) For ar > 0 and S G B^'1),

nP(|X| > anz,X/|X| G S) = nP(a"1X G K,s)

if //(öV^s) = 0. Since, by Theorem 1.14, p has the scaling property

p(xB) = x~ap(B) it follows by Theorem 1.8 that p(dVX:s) = p(Vx,os)

for every x > 0 and S G B(Sd 1). Hence, with an = (yu(V1)gd-i))1/a;an,

nP(|X| >anz,X/|X| G • ) ^y x~aa(-) onß^"1),

where cr(-) = p(Vi,.)/p(Vi^d-i) is a probability measure on 23(§d_1).For w > ai, let n = n(n) be the largest integer with an < u. For any

x > 0 and S G #(§d_1) we have

nP(|X| >qn+ix,X/|X| G 5) P(|X| >us,X/|X| £ S)

nP(|X| >an)-

P(|X| > u)

nP(|X| >qn£,X/|X| G S)

nP(|X|>an+1)

Moreover, if a(dS) = 0, then

nP(|X|>an+1a;,X/|X|G5)=

n (n + 1) Pff-^X G K,s)

nP(|X|>an)~

n + 1 nF(a^X G 7i^-i)

as u —y oo. Similarly for the upper bound. Hence (ii) holds. D

An immediate consequence of Theorem 1.15 is the following result (seethe proof of Theorem 1.15 for details).

Corollary 1.20 Let \-\a and \ • \b be two norms on Md and let X be

an Md -valued random vector. Then X is regularly varying with index a

with respect to the norm \ • \a if and only z/X is regularly varying with

index a with respect to the norm \ • \b-

It is clear that the corresponding spectral measures do not coincide for

different norms. See Chapter 5 for explicit examples and interpretations

of how the choice of norm affects the spectral measure.

Let us now consider another formulation of multivariate regular vari¬

ation which we will use to show that a random vector with regularly

varying components need not be regularly varying.

Theorem 1.21 Let X be an Md-valued random vector. Then the fol¬

lowing statements are equivalent.

—d

(i) There exist a relatively compact set E G B(M \{0}) and a nonzero

Radon measure p on B(M \{0}) such that, as u —y oo;

P(XGW-)^M-) onB(Md\{0}). (1.3)(X G uE)

(ii) There exist an a > 0 and a probability measure a on B(Sd x) such

that, for every x > 0, as u —> oo,

x|>^,x/|x|g_o^_m>) onB(Sd-i}<P(|X| > u)

Remark 1.22 If (i) holds, then by essentially the same arguments as

in the proof of Theorem 1.14 there exists an a > 0 such that p(xB) =

x~ap(B) for every x > 0 and B G #(Rd\{0}) and (ii) holds with the

same a. If (ii) holds, then (i) holds and p satisfies the scaling property

above with the same a (see the proof of Theorem 1.15 for details).

Remark 1.23 Note that if (1.3) holds, then

p(xe«,:)=P(X6«-)P(xeug)AifH onB^X{0})P(X e »£) P(X e uE) p(x 6 uE) ß{E)

~ —d ~ ~

for any relatively compact E G B(M \{0}) with p(dE) = 0 and p(E) >

0.

Proof, (ii) =^> (i) Follows by the same arguments as in the proof of

Theorem 1.15.

(i) => (ii) The limit measure p has the scaling property described in

Theorem 1.14 (this can be shown by essentially the same arguments as

in the proof of Theorem 1.14). Hence, by Theorem 1.8, p(dVi^d-i) = 0

and p(dVU:s) = p(VUjos) for every u > 0 and S G B(Sd~1). Hence, if

MK,ös) = 0, then

X| > ux, X/|X| G S)_

P(X G uVXis) P(X G uE)

X| > u) P(X G uE) P(X G nViiSd-i)

^KV*,s)

=x-aKVi,s)

M(^i,sd-i) v(Vi^d-i)

as u —y oo; i.e. (ii) holds with a(-) = p(V\,.)/p(Vi^d-i). D

The following example shows that a random vector with regularly vary¬

ing components need not be regularly varying.

Example 1.24 Let Q be a probability measure on [0, l]2 such that for

every integer n > 1, Q assigns mass 2~n, uniformly distributed, to the

line segment between (1 - 2~n, 1 - 2~n+1) and (1 - 2~n+1,1 - 2~n).Define the distribution function C by C(u\, U2) — Q([0, u\\ x [0, U2]) for

ui,U2 G [0,1]. Note that C(wi,0) = 0 = C(0,U2), C(ui,l) = u\ and

C(1,W2) = U21 i.e. C is a so-called copula (see e.g. Joe [25]). Note also

that for every n > 1 (with C(u, u) = 1 — 2u + C(u, u))

C(l - 2~n+\ 1 - 2-n+1)/2~n+1 = 1

and

Ü(l - 3/2n+1,1 - 3/2n+1)/(3/2n+1) = 2/3.

In particular lim^i C(u,u)/(1 — u) does not exist. Take a > 0 and let

F be given by F(x) = 1 — x~a for x > 1. Finally, let X be an R2-valued

random vector with distribution function given by ¥(X^ < x\, X^1' <

X2) = C(F(xi),F(x2))- Since X^ and X^ have distribution function

F, it follows by Corollary 1.18 that they are regularly varying. However,

P(XeF-i(«)((l,oo)xR))-6("',')/(l «), (1-4)

which as we have seen does not have a limit as u 11 •If X were regularly

varying with some limit measure p, then by Theorem 1.8 (1, 00) x (1, 00)and (1, 00) x R would be relatively compact //-continuity sets of positive

//-measure and thus, by Theorem 1.21 and Remark 1.23, the left-hand

side of (1.4) would converge as u t 1- We conclude that X is not

regularly varying.

Let us now consider yet another possible formulation of multivariate

regular variation; there exists an a > 0 and a slowly varying function

L (L is a strictly positive Lebesgue measureable function on (0, 00)


satisfying lim^^oo L(ux)/L(u) = 1 for every x > 0) such that for every

x G Rd\{0}

,.P«x,X) >u) . .

. n . n .

hm —--— = wix.) exists and is finite (1.5)

u^oo U~aL(u)W V ^

and w(x) = 0 is possible for some but not all x G Rd\{0}. It follows

immediately that the function w is homogeneous, w(ux.) = waw;(x) for

every u > 0 and x G Rd\{0}. It is easy to show that (1.2) implies

(1.5). In Basrak, Davis and Mikosch [4] the following theorem proves

equivalence of (1.2) and (1.5) under some additional assumptions.

Theorem 1.25 LetX. be an Md -valued random vector. Then (1.2) and

(1.5) are equivalent if either (i) a is a positive noninteger or (ii) X has

nonnegative components and a is an odd positive integer.

Let X be a random vector satisfying (1.2), i.e. X is regularly varying.

Then X satisfies (1.5) and the limit function w is uniquely determined

by a and a. On the other hand, the limit function w determines a but

not necessarily the spectral measure a if a. > 0 is a positive integer.

Consider the following example.

Example 1.26 Fix an integer a > 1. We will construct two regularly

varying random vectors Xi and X2 with tail index a and different

spectral measures P((cosOi, sinOi) G • ) and P((cos02, sin82) G • ),such that the limit functions w\ and u>2 in (1-5) coincide. Let Oi be

a [0, 27r)-valued random variable with density function fi(9) > w for

9 G [0, 2tt) and some w > 0. Take v G (0, w) and let O2 have density

function /2 given by

hiß) = fi(0) + vsin((a + 2)9), 9 G [0, 2tt).

Let R ~ Pareto(a), i.e. ¥(R > x) = x~a for x > 1, be independent of

Qi, i = 1, 2, and put

A/ RcosGi \

'~ I RsinGi J'


Take x G R2\{0} and let ß G [0, 2tt) be given by

x_

( cosß

Ixl I sin/5

Then, for u > |x|

x,X2> >u)-P((x,Xi) >u)

= P((x/|x|, X2) > u/|x|) - P«x/|x|, Xi) > u/|x|)/»oo /»/3+arccos((u/|x|)/r)

= v / ar-a-1sin((o; + 2)^)d^drJu/\x\ Jß—arccos((u/|x|)/r)

/»oo />/3+arccos((u/|x|)/r)

= v ar'-1 sin((a; + 2)0))d0drJu/\x\ Jß—arccos((u/|x|)/r)

/»oova r _a_x

'u x

r"a_i cos((a + 2)(ß + arccos((w/|x|)/r)))

— cos((a + 2)(ß — arccos((u/|x|)/r))) jdr/»OO

sin((a + 2)ß) / r~a_1 sin((a + 2) arccos((u/|x|)/r))drAt/lxl

2t>Q!

OL -\- L./u/lx

Using standard variable substitutions and trigonometric formulas the

integral can be rewritten as follows:

/»OO

/ T»-«-1 sin((a + 2) arccos((w/|x|)/r))drAt/|x|

ra_1 sin((a; + 2) arccos(r))dr-a I

i———/ cosa_1(r) sin((a + 2)r) sin(r)dr

lxla Jo

u-a rir/2'

cosa_1(r) cos((a + l)r)dr0

/»TT/2cosa(r) cos((q! + 2)r)dr.

x-a

u-ar/2

-a I

JO

The two last integrals are zero for every a G (0, oo); these integrals can

be found in Gradshteyn and Ryzhik [19] p. 392. Hence, for u > |x|,

P((x, X2) >u)= P((x, Xi> >u). m

Theorem 1.25 (i) is proved in Basrak, Davis and Mikosch [4] by show¬

ing that the limit function w in (1.5) uniquely determines the spectral

measure a in (1.2) if a > 0 is not an integer. The above example shows

that the idea behind the proof can not be extended to integer-valued

tail indices. Whether the result still holds in this case is not known.

1.4 Sums of regularly varying random vec¬

tors

In this section we will derive some useful results concerning sums of

regularly varying random vectors. The results generalize known results

in the univariate case to the multivariate setting but the techniques used

in the proofs are quite different. Let us start with a result on linear

transformations of a regularly varying random vector (see also Basrak,

Davis and Mikosch [3]). Note that the assumption on X in Theorems

1.27 and 1.28 is weaker than that of multivariate regular variation since

we do not require that the limit measure is nonzero.

Theorem 1.27 Let X be an Md-valued random vector and suppose

there exist a sequence (an), 0 < an | oo, and a Radon measure p

on ß(Rd\{0}) with /i(Rd\Rd) = 0 such that nP(a~lX e ) A p(-) on

B(M^\{0}). If T : Md -+ MP, p < d, is a linear transformation of full

rank, then

nF(a~1T(X) G •) ^y poT'^ DMP) on B(M?\{0}). (1.6)

Proof. Let B e B(m7\{0}) be relatively compact with p o T~1(dB n

MP) = 0. If B C RP\RP, then

n¥(a~1T(X) £ B) = p o T~X(B n Mp) = 0

for all n. Hence we can without loss of generality assume that B D

MP =£ 0. Since B is relatively compact there exists an e > 0 such that

infxGßnKp |x| > e. For x G B fl Rp take y such that T(y) = x (since

1.4. Sums of regularly varying vectors 29

T is onto such a y exists). Since |T(y)| < ||T|||y| and ||T|| < oo for

all linear transformations T : Md -> MP, |y| > |x|/||T|| > e/||T|| for

all y G T~l(B n Rp). Hence T~l(B n MP) G #(Rd\{0}) is relatively

compact. Since T is continuous and p(M \Md) = 0,

0 = p(T~l(dB n Rp)) = p(T~l(d(B n Mp) n MP)) = p(dT~l(B n MP))

and hence

nF(a~lT(X) G B) = nF(a~1T(X) G B n Rp)

= nP(T(a"1X) £5nlp)

= nP(a"1XGT-1(SnRp))

->• //(t-^sdrp))

as n —y oo, from which the conclusion follows. D

We proceed by considering sums of a fixed number of independent reg¬

ularly varying random vectors.

Theorem 1.28 Let X be an Md-valued random vector and suppose that

there exist a sequence (an), 0 < an t oo, and a Radon measure p on

B(M^\{0}) with p(Md\Md) = 0 such that nP(a"1X G • ) ^ p(-) on

£(rA{0}).

(i) If X is a random vector in Md, independent of X, and if there

exists a Radon measure p on B(M \{0}) with p(M \Md) = 0 such

that nP(a"1X G • ) A- p(-) on #(Rd\{0}); then

nF(a~1(X + X) G • ) -A p(-) + p(-) on B(M^\{0}).

(ii) If for some k, there exist independent and identically distributed

random vectors Xi,..., X& such that X = Xi + • • • + X&, then

t

nP(a"1Xi G • ) A -p(-) on B(m\{0}).K

Remark 1.29 A statement similar to (i), for Revalued random vec¬

tors, is proved in Resnick [33] (Proposition 4.1 p. 85).

Proof, (i) Take t\ > 0 and 2 > 0 and note that, by Theorems 1.8 and

1.14, p(d(Bc0jJ) = p(d(B^J) = 0. Since

nF(a-1(X,X)eBc0^ixBc0je2)= nF(a~1X G Bgjei) P(a"1X G Bg>ea) - 0,

as n ->• 00, it follows that nP(a"1(X,X) G • ) -A p(-) on #(R2d\{0}),where p is a Radon measure which concentrates on ({0} x Md) U (Md x

{0}). Let T : R2d - Rd be the linear transformation T(x, x) = x + x.

By Theorem 1.27,

nP(a"1(X + X) G •) ^/2°^_1(-nRd) on #(Rd\{0}).

Hence for any B G #(Rd\{0});

poT~l(BnMd) = £({(x,x) :x + ÏG£nRd})= £({(x,0) :x + 0G5nRd})

+£({(0,x) :0 + ÏGSnRd})= /z(S)+£(S).

(ii) Since nF(a~1X G • ) A //(•) on #(Rd\{0}) it follows that for any

subsequence (nj) such that nj —y 00 as j —y 00, n^ P(a~1X G • ) A p(-)

on #(R \{0}). Hence, it follows by (i) that any subsequential vague

limit p\ of (nF(a~lXi G • )) must satisfy p\ = p/k. Hence, we only

need to show that {nF(a~1X\ G • )} is relatively compact in the vague

topology. By Theorem 1.4, {nF(a~1Xi G • )} is relatively compact in

the vague topology if and only if supn>1 nF(a~1Xi G B) < 00 for every

relatively compact B G B(M \{0}). We prove this by contradiction.

Suppose that there exists a relatively compact set B G B(M \{0}) such

that supn>1nF(a~1Xi G B) = 00. Then there exists an r > 0 such

that supn>1 nF(a~1Xi G Bq r) = 00. Since nF(a~1Xi G Bq r) < 00

for every n this implies that limsupn^.00 nF(a~lXi G Bq r) = 00. Take

e G (0,r/k). Then

P(a"1X G Bc0jr_ke)= F(a-1(Xl + ... + Xk)eBc0,r_ke)> F(a~1X1 G Bc^r, a~lX3 G B0,e for j = 2,..., fc)

= P(a"1X1 G ^jr)P(a"1X1 G ßo.e)*-1.

Hence

limsupnP(a~1X G 5o,r-fce)

> limsupnP(a-1Xi G 5£;T.) P^Xi G ßo^)^"1 = oo.

n—>-oo

This contradicts the assumption that {nP(a~1X G • )} is relatively

compact and we conclude that {nF(a~1Xi G • )} must be relatively

compact in the vague topology. D

Above we considered a sum of a fixed number of terms. Let us now

consider the case with a random number of terms N, where N is inde¬

pendent of the terms Xk.

Theorem 1.30 Let (Xk)k>i be a sequence of independent and identi¬

cally distributed Md -valued random vectors and suppose that there exist

a sequence (an), 0 < an t oo, and a nonzero Radon measure p on

B(Md\{0}) with p(Md\Md) = 0 such that nF(a~lX1 G • ) A p(-) on

B(M \{0}). If N is a nonnegative random variable with X^Lo^C-^ =

n)(l + e)n < oo for some e > 0 and N is independent of (Xk)k>i, then

N

nF(a-1J2xk -)AE(N)p(-) onB(Md\{0}).k=i

If (Yk)k>i is another sequence of independent and identically distributed

Md-valued random vectors, independent of N, such that nF(a~1Yi G

• ) A- 0 as n —y oo, then nF(a~l Y2k=i ^k £ ' ) A 0 as n ^ oo.

Proof. Take a relatively compact B G #(Rd\{0}) with p(dB) = 0.

Since for all n, nF(a~1 J2k=i Xfc e M.d\Md) = E(N)p(M.d\Md) = 0, we

may without loss of generality assume that B cMd. Let 7 = infx_e |x|.Since

N

^XfcGB)n

k=l

oo

= ^nP(a-1^X,Gß)P(AT = 0

00 £

< ^nPfc1 ]T |Xfc| > 7) F(AT = 0 (1.7)1=1 k=i

and |Xfe| is univariate regularly varying, it follows from Theorem 3

in Embrechts, Goldie and Veraverbeke [15] that the right-hand side of

(1.7) converges to ~E(N)p,(Bg ). Hence, using Pratt's Theorem (seePratt [31]) and Theorem 1.28, we conclude that we may interchange

the sum and the limit to obtain

N oo I

lim n F(a~1 V Xk G B) = lim V n P«1 V Xfc G B) F(N = I)n—>-oo ^-—' n—>-oo *-—' ^-—'

fc = l / = 1 fc = l

00 I

= V lim nF(a-1J2xkeB)F(N = l)

= E(N)p(B)

n—>-oo

Z=l fc=l

for every relatively compact B G ß(R \{0}) with p(dB) = 0. The

second claim is proved similarly. D

Remark 1.31 The moment condition on N, J2=o^(N = n)(l + e)n <

00 for some e > 0, comes from the remark following Theorem 3 in

Embrechts, Goldie and Veraverbeke [15]. This result concerns random

sums of univariate random variables with subexponential tails. Since

we use this result for terms which are regularly varying the moment

condition can in fact be substantially weakened.

The following lemma can be used for instance in connection with The¬

orem 1.28 to show that if we add any random vector with all moments

finite to an independent regularly varying random vector X with limit

measure p, then the sum is regularly varying with the same limit mea¬

sure p.

Lemma 1.32 Let X be an Md-valued random vector. J/E(|X|m) < oo

for every m G N, then for every regularly varying sequence (an), 0 <

an t oo, and every relatively compact set B G B(M \{0}), nP(a~1X G

B) —y 0 as n —y oo.

Proof. Fix a > 0 and let (an), 0 < an î oo, be a regularly varying

sequence with index 1/a. Let B G B(M \{0}) be relatively compact.

Then there exists an x > 0 such that B C Bq x. Hence, for n large,

nP(a^X G B) < nF(a~1X G 5£J = nP(|X| > anx).

Define / by f(t) = inf{n G N : an > t}. Then by Theorem 1.5.12 p. 28

in Bingham, Goldie and Teugels [7] / is regularly varying with index a,

i.e. f(t) = taL(t) for some slowly varying function L, and f(an) ~nas

n —y oo. Hence

nP(|X| > anx) ~ /(an)P(|X| > anx) as n —y oo.

By Markov's inequality we have P(|X| > tx) < E(|X|m)/(te)m < oo for

every t > 0, x > 0 and m > 0. Hence, for every ra > 0,

/(an)P(|X| > anx) < aZL(an)a-mx-mE(\X\m) = C(m)al-mL(an).

Taking m > a and letting n —>• oo yields nP(|X| > ana:) -^ 0 from

which the conclusion follows. D

An immediate consequence is the following result.

Corollary 1.33 Let X be an Md-valued regularly varying random vector

with tail index a > 0 and spectral measure a with respect to the norm

| • |, and let b G Md be a constant vector. Then X+ b is regularly varying

with the same tail index and the same spectral measure with respect to

the norm I • I.

Chapter 2

Regular variation for

multivariate additive

processes

Stochastic processes with heavy-tailed marginal distributions have be¬

come increasingly important in many applications such as communica¬

tion networks, hydrology, insurance mathematics and mathematical fi¬

nance. Many interesting examples can be found in the collections Adler,

Feldman and Taqqu [1] and Rachev [32]. A lot of effort has been put

into studying such processes and into finding the tail asymptotics of the

distribution of functionals of the processes (see e.g. Embrechts, Goldie

and Veraverbeke [15], Rosihski and Samorodnitsky [36] and Braverman,

Mikosch and Samorodnitsky [9]). However, whereas problems concern¬

ing the tail behavior of univariate stochastic processes have been studied

successfully for quite some time, multivariate processes have received

far less attention. In this chapter we focus on multivariate additive

processes, i.e. stochastically continuous processes with independent in¬

crements, which at some fixed time t > 0 are regularly varying, and

we study the tail behavior of vectors of functionals acting on such pro-

35

36 Chapter 2. Regular variation for additive processes

cesses. By tail behavior we mean a regular variation limit measure. The

basic intuition underlying all the results is the following: the process

reaches a set far away from the origin by making one big jump at some

time t < t, and in comparison to the size of the jump, the process does

not move much before r nor between r and t.

We prove tail equivalence between the distribution of Xt and its associ¬

ated Levy measure vt in the sense that if pt is a nonzero Radon measure

on #(Rd\{0}) with ^(Rd\Rd) = 0, then

nP(a"1X, G • ) A pt(-) on £(Rd\{0}) (2.1)

if and only if

niyt(an-) A pt(-) on B(m\{0}). (2.2)

This is a multivariate version, in the regularly varying case, of a result

in Embrechts, Goldie and Veraverbeke [15] which says that

F(Xt > x) ~ vt({y e R : y > x}) as x - oo,

for Xt subexponential. Moreover, we determine the implications of

regular variation of Xt on the joint tail behavior of vectors of functionals

acting on the underlying process (Xs)s>q. For example, we study the

componentwise suprema

X,* = (sup Xi1),..., sup X^)0<s<t 0<s<t

and the componentwise suprema of the jumps

XiA = (sup AX,..., sup AX^),0<s<t 0<s<t

and we show that if the additive process (Xs)s>0 satisfies the regular

variation condition (2.1), then

nF(a~1X*t e-)A pt(-) and nP(a"1Xf £-)A pt(-)

on #(R+\{0}), i.e. we have convergence to the same limit measure. In

this case, knowing the joint tail behavior of X^ is enough to determine

the joint tail behavior of X£ and XA respectively. However, there are

37

many other interesting choices of functionals for which the joint tail

behavior cannot be determined by that of Xt alone. Typically, we would

need information of the type: the probability that Xs for some s G [0, t]reaches a set anBs, where Bs is allowed to vary with s. This can be

d

formulated in terms of a vague limit on B([0, t] x (R \{0})) for the graph

{(s, Xs) : 0 < s < t} of our process. We prove tail equivalence between

the graph of the process and the associated measure v, where v([0, s] x

B) = vs(B) for s G [0, t] and B G #(Rd\{0}), in the sense of having the

same vague limit on #([0,£] x (R \{0})). Since the precise formulation

of the result (Theorem 2.11) requires some additional notation and a

few technicalities, we refrain from stating it at this point. Using this

result we can determine the joint tail behavior of the componentwise

integrals

•I X^ds,...o

in the sense of the vague limit of nF(an1It G • ) on B(M \{0}). As a

special case, if (Xs)s>o is a Levy process, then it follows that

nP«1!, e-)A^-p(-) onB(Md\{0}),a + V

where p and a > 0 are such that Pt(-) = tp,(-) and p(u •) = u ap(-) for

u>0.

The chapter is organized as follows. We begin in Section 2.1 by intro¬

ducing additive processes and recall some of their properties. In Section

2.2 we prove the main results on tail equivalence and joint tail behavior

of functionals acting on the components of the processes.

The symbol ~ denotes both asymptotic equivalence, i.e. f(x) ~ g(x)as x —y oo if linx^oo f(x)/g(x) = 1, and that a random vector has

a certain probability distribution, i.e. X ~ F means that X has the

distribution F. The dual use of ~ should not cause any confusion.

2.1 Additive processes

In this section we introduce additive processes following Sato [38]. An

additive process (Xt)t>o on Md is a stochastically continuous stochastic

process with independent increments, starting at zero. There exists a

version of it which has right-continuous sample paths with left limits.

We will always choose such a version. If in addition (Xt)t>o has station¬

ary increments, then it is called a Levy process. For an additive process

(Xt)t>o on Md, for every t, the distribution of X^ is infinitely divisible,

i.e. for any positive integer n there exist independent and identically dis¬

tributed random vectors Zi;ri)i,..., ZnjT1)i such that Xt = Y^i-i ^i,n,t-If (Xt)t>0 is a Levy process, then we can take Zijn>i = Xti/n-Xf(i_1)/n.

We denote by F the characteristic function of a probability distribution

F onRd,

F(z) = / e*<z'x)F(dx).

For any infinitely divisible probability distribution F on Md we have the

Lévy-Khintchine representation (Sato [38] Theorem 8.1 p. 37):

F(z) = exp ( - i<z, Az) + z<7,z> (2.3)

+ / (eiM ~ 1 - t<z,x)l{x:|x|<1}(x))i/(dx)Y</R<*\{0}

J

z G Md, where A is a symmetric nonnegative definite d x d matrix, v is

a measure on Rd\{0} satisfying

/ (|x|2Al)i/(dx) < oo, (2.4)jRd\{0}

and 7 G Md. We call (A, v,~{) the generating triplet of F. The matrix

A and the measure v are called, respectively, the Gaussian covariance

matrix and the Levy measure of F. When A = 0, F is called purely

non-Gaussian.

An important result for additive processes is the Lévy-Itô decomposition

2.1. Additive processes 39

which we will recall below. First some notation. Let

D0)6 = {x G Rd : a < |x| < b}, for 0 < a 0 be an

additive process on Md defined on a probability space (Q, T, F) with sys¬

tem of generating triplets ((At, vt,~ft))t>o and define the measure v

on (0,oo) x (Rd\{0}) by u((0,t] x B) = ut(B) for B G #(Rd\{0}).Let Qq G T, P(Oq) = 1, be such that t \-y Xt(u;) is right-continuous

in t > 0 with left limits in t > 0 for each u G Q0 and define, for

H eB((0,oo) x (Rd\{0}));

\ 0, foruj^üo-

Then the following hold.

(i) {£,(H) : H G B((0, oo) x (Rd\{0}))} is a Poisson random measure

on (0,oo) x (Rd\{0}) with intensity measure v.

(ii) There exists Q\ G T with F{£l\) = 1 such that, for any to G Q\,

XKlü) = lim / {x£(d(5,x),ü;) -xï7(d(s,x))} (2.6)

+ / xf(d(s,x),u;)J(0,t]xDltOO

is defined for all t G [0, oo) and the convergence is uniform in t

on any bounded interval. The process (X^) is an additive process

on Md with ((0,z^,0)) as the system of generating triplets.

(iii) Define

X?(u;) = Xt(uj) - X](uj) for ooeüi.

There exists Q2 £ J7 with P(f22) = 1 such that, for any u G f22,

X\(lS) is continuous in t. The process (Xf) is an additive process

on Md with ((^,0,7^)) as the system of generating triplets.

(iv) The two processes (Xj) and (Xf) are independent.

In connection to this theorem we give the supplementary result which

says that the part of the additive process containing the small jumps

will always have finite moments of all orders. This will be relevant when

studying the tails of a regularly varying additive process.

Lemma 2.2 Let Yt = lime^o J(0t]xD {x£(d(s,x)) —xi/(d(s,x))}. Then

for every t > 0 and m G N, E(|Yf |m) < oo.

Proof. Fix arbitrary t > 0 and integer m > 1 (the case m = 0 is

trivial). For notational convenience, let Zj, Xj and Ytj denote the jth

component of z, x and Y^ respectively. Since any two norms on Md

are equivalent we may without loss of generality take | • | to be the

standard Euclidean norm, i.e. the norm given by |x| = (52i=1x2)1'2.Note that by the Lévy-Khintchine representation for infinitely divisible

distributions

j (ei<z,x> _ x _ ^x^^dx) and f |x|V(dx)

exist finitely. Let Yf = f(0t-\xD (x£(d(s,x)) — xî/(d(s,x))}. Then

Yf A- Yt as e \. 0 and hence the characteristic functions converge,

E(e*<z'Y*>) = exp{ j (e*<z'x) - 1 - i(z,x»i/t(dx)}

-* exp{ j (ei(z'x)-l-z(z,x))^(dx)} =E(e^z'Yt)).

If n\,..., nd G N with rij > 1 for some j G {1,..., d}, then

/ x2ni •

...

• x2dndvt(dx) < f xf-3vt(àx) < j |x|2^(dx) < oo.

Jdoa Jd0i1 Jd0i1

Moreover,

E(|Y<I2"*) = J2 „,"*'„ ^Y*T rM").ft ]_ • • • • 't'c/ •

where since nj > 1 for some j,

= f x\ni •...•x2dndvt(d*) <oo.

Jd0>1

2.2. Regular variation of functionals 41

Hence E(|Yi|2m) < oo. However, E(|Yi|m) < (E(\Yt\2m))1/2 from

which the conclusion follows. D

2.2 Regular variation for multivariate ad¬

ditive processes and for vectors of func¬

tionals acting on such processes

We now turn to the main topic of this chapter; the tail behavior of

multivariate additive processes and functionals of them. In the uni¬

variate case it is well known (see e.g. Embrechts, Goldie and Veraver¬

beke [15]) that for an infinitely divisible regularly varying (or even

subexponential) random variable X with Levy measure v it holds that

F(X > u) ~ v({x : x > u}) as u —y oo. This property is sometimes

referred to as tail equivalence of X and v. For a univariate additive

process (Xt)t>o with system of generating triplets ((At, Vf>lt))t>o this

implies that if Xt is regularly varying (subexponential) for some t > 0,

then F(Xt > u) ~ Vt({x '• x > u}) as u —y oo. The intuition behind

this result is explained by what is sometimes referred to as the "large

deviations principle" : unlikely events happen in the most likely way. In

this case this is interpreted as follows. The most likely way the process

becomes large at time t is due to one big jump of the process before

time t. The Levy measure of {x : x > u] gives the intensity of jumps to

this set during (0, t] and the probability of exactly one jump to this set

is asymptotically ut({x : x > u}). The same intuition holds also in the

multivariate case as the next result shows. The tail equivalence should

now be interpreted as equality of the limiting measures associated with

multivariate regular variation.

Theorem 2.3 Let (Xt)t>o be an additive process on Md with system of

generating triplets ((At, ^ti1t))t>o- Fix an arbitrary t > 0. Then the

following statements are equivalent.

(i) There exist a sequence (an), 0 < an f oo, and a nonzero Radon

measure pt on B(M \{0}) with pt(M \Md) = 0 such that

nF(a~1Xt G • ) A pt(-) on B(M~f\{0}). (2.7)

fzz) There exist a sequence (an), 0 < an f oo, and a nonzero Radon

measure pt on B(M \{0}) wz£/& pt(M \Md) = 0 swc/i that

nut(an-) ^y pt(-) on B(Md\{0}). (2.8)

Furthermore, the sequences (an) and the measures pt in (i) and (ii) can

be taken to be equal.

Remark 2.4 (i) Note that for any t > 0 and any infinitely divisible

random vector Y there exists an additive process (Xs)s>0 such that

Y = Xt. Hence Theorem 2.3 can be reformulated in terms of infinitely

divisible random vectors.

(ii) Note also that if Theorem 1.25 (i) was true for arbitrary tails indices

a > 0, then Theorem 2.3 would be easily proved using the results in

Embrechts, Goldie and Veraverbeke [15] for univariate subexponential

infinitely divisible random variables.

Proof of Theorem 2.3. To prove this theorem we will make use

of the Lévy-Itô decomposition, Theorem 2.1, which says that X^ has

representation

Xt=Yt+Jt + X2 a.s,

where Yt, Jt and X2 are independent and, with the notation of Theorem

2.1,

Yt = lim/ {x£(d(S,x))-x£(d(S,x))},

Jt = [ x£(d(s,x)),J(0,t]xDl!OO

Xt = Xt — Yt — Jt •

X2 is Gaussian and hence has finite moments of all orders. By Lemma

2.2, Yt has finite moments of all orders and will therefore, as we will

see, not contribute to Xt being regularly varying. It will be sufficient

to consider the part Jt being the accumulated big jumps of the process

up to time t. Since £ is a Poisson random measure, the characteristic

function of Jt is

E(ei(z'J*>) = exp { / (e*<z'x) - l)^(dx))

and hence Jt has representation as a compound Poisson random vector

Nt

Jt = / ,Jk,t,k=0

where Nt ~ ~Po(vt(Dij00)), i.e. Nt is Poisson distributed with parameter

vt(DijOQ), J0,t = 0, Jk,t ~ vt( • n Dii00)/vt(Dii00) and all vectors are

independent.

(i) =>• (ii) To prove this implication we will show that

nF(a-1Jhte-)Apt(-)/ut(D^00) as n ^ oo. (2.9)

Then the implication follows since Jk,t ~ vt( • n.Di)00)/i^(.Di)00) and for

any relatively compact B G B(M \{0}) and large enough n, vt({anB} D

-Di,oo) = ut(anB). So if (2.9) holds, then also

nut(an- ) A- pt(-) as n —y oo.

The proof goes in two steps. First we show that for (2.7) to hold it is

necessary that

nF(a~1Jt G • ) A pt(-) as n ^ oo, (2.10)

and for (2.10) to hold it is necessary that (2.9) holds. We start by show¬

ing that the set {nF(a~1Jt G • )} is relatively compact in the vague

topology. As in the proof of Theorem 1.28 we prove this by contradic¬

tion. Assume that there exists a relatively compact B G B(M \{0}) such

that supn>1 nF(a~1Jt G B) = oo. Then there exists an r > 0 such that

B C Bq rand hence supn>1nF(a~1Jt G Bq r) = oo. Since nF(a~1Jt G

Bqt) is finite for every n this implies limsupn_>.00 nF(a~1Jt G Bq r) =


oo. Take e G (0,r/3). Then

lim sup nF(a~1Xt G #o,r-2e)n—>-oo

= lim sup nP^"1^ + Jt + Xf2) G ßg>r_2e)n—>-oo

> lim sup nP«1J, G 50c,r)HWYt G A),e)P«%2 e ßo,e)n—>-oo

= OO.

Thus {nF(a~1Xt G • )} is not relatively compact which is a contradic¬

tion. We conclude that {nF(a~1Jt G • )} is relatively compact. Let

(ni) be a subsequence such that lim^oo rii = oo and let pij be the

vague limit of (n(F(a~1Jt G • )) as i —)• oo. Since Yt and X2 have fi¬

nite moments of all orders, by Lemma 1.32, for every relatively compact

BeB(Md\{0}),

ni P(a~1Yi eß)40 and m F(a~^X2 G B) -> 0 as i -+ oo.

Then, by Theorem 1.28 (i), niP(a~1Xt G • ) A //M(-) as i ^ oo.

However, we have assumed that (2.7) holds so p,\j = Pt- Hence, (2.10)holds.

We continue with a similar argument to show that (2.9) holds. Let us

start by showing that the set {nF(a~lJ\j G • )} is relatively compact in

the vague topology. Assume that there exists a relatively compact B G

d

B(M \{0}) such that supn>1 nF(a~ J\,t G -ß) = oo. Then there exists

an r > 0 such that B C B^r and hence supn>1 n P(a~1Ji,t G Bq r) = oo.

Since nF(a~1Jijt G Bq r) is finite for every n, limsup^.^ nF(a~1Jiit G

Bq r) = oo. We have

limsupnP^Ji G B^r) > limsupnP(a-1Ji,t G B^r)F(Nt = 1) = oo.

Thus {nF(a~lJt G • )} is not relatively compact which is a contradic¬

tion. We conclude that {nF(a~lJ\j G • )} is relatively compact. Let

(nj) be a subsequence such that lim^oo nj = oo and let p2,t be the

vague limit of (ujF(a~lJ\j G • )) as j —y oo. By Theorem 1.30 it

follows that

n, PK/J* e ) A E(Nt)p2,t(-) = vt(Di,oo)p2,t(-) as j ^ oo


and hence we must have p2,t = Pt/^t(Di^oo)- Hence, (2.9) holds.

(ii) => (i) Since Jk,t ~ vt( • D Dij00)/vt(Dij00) and we have assumed

that nvt(an- ) A- pt(-) as n —y oo, it follows that

nP(a~1Jfc;i G • ) A- pt(-)/vt(Di,oo) as n ^ oo

since for any relatively compact B G B(M \{0}) and large enough n,

fi({an.B} n -Di,oo) = vt(anB). Then, by Theorem 1.30,

nPK1^ G • ) ^ E(^)M*(-)M(^i,oo) = Mt(-) as n ^ oo.

Now, since Xt = Y* + J^ + X2 a.s. where the terms are independent, Y*

and X^ have finite moments of all orders and nF(a~1Jt G • ) -A pt(-) as

n —y oo, the conclusion follows by combining Lemma 1.32 and the first

part of Theorem 1.28. D

In this chapter we focus on additive processes. However, in order to

avoid having to prove essentially the same result twice (also in the fol¬

lowing chapter) we formulate Theorem 2.7 below for strong Markov pro¬

cesses. We first show that additive processes are indeed strong Markov

processes (see Remark 2.6 below for details on what is meant by strong

Markov process).

Theorem 2.5 An additive process (Xt)t>o on Md is a strong Markov

process.

Proof. By Remark 1 following Theorem 7 p. 61 in Gihman and Sko-

rohod [21] (and as seen from the proof of Theorem 7) a Markov process

(Yt)t>o on Md is strong Markov if it has a right-continuous version and

if for any bounded continuous / and t > 0 it holds that F defined by

F(s,x) = Es^(f(Yt)) is continuous at (s,x) for any (s,x) G [0,t) x Md.

For an additive process (Xt)t>0 on Md we have (see e.g. Sato [38] p. 56

and 57)

F(S,x) = E*'x(/(Xf))

= J /(y)i^(x, dy) = J /(x + yK*(dy),

where p.sj denotes the distribution of Xt — Xs. Fix arbitrary x G Md,

s,t G R+ with s < t and a bounded continuous /. Consider sequences

(sn) and (xn) such that sn —y s and xn —y x. Define, for y G Md and

Bend,

fn(y) -/(x„ + y), /oo(y) = /(x + y),

pn(B) = pSn,t(B), Poo(B) = ps,t(B),

and note that fn —> /oo pointwise and /in -^ //qo- Let E be the set of

y such that fn(yn) —> Zoo(y) fails to hold for some sequence (yn) con¬

verging to y. By Theorem 5.5 p. 34 in Billingsley [5], pn -A p^ implies

Mn ° /n_1 ^ Moo ° /oo1 if Poo(E) = 0. Since fn converges to /^ uni¬

formly on compact sets and since /oo is continuous, E is empty. Hence

the hypothesis of Theorem 5.5 is satisfied. Introduce random variables

Xn and Xoo such that Xn ~ pn o f-1 and X^ ^^0 /-1. Since /

is bounded, {fn} is uniformly bounded and hence {Xn} is uniformly

integrable. Finally, by Theorem 5.4 p. 32 in Billingsley [5] and upon

transformation of integrals,

J /„(z)/zn(dz) = J zpnof-1(dz)=E(Xn)

-y E(XOQ) = / zpoo of~1(dz) = / /oo(z)^oo(dz).

Hence, F is continuous at (s,x) for any (s,x) G [0,t) xMd. D

Remark 2.6 By strong Markov process we mean a Markov process

which satisfies Definition 2 p. 56 in Gihman and Skorohod [21]. In

particular, a strong Markov process is not necessarily temporally ho¬

mogeneous. Note that essentially only one property of strong Markov

processes is used in this and the following chapter, namely that

E(l{r<i}l{xt-xTe^})=E(l{r<t}E^x^(l{Xt-xTeßoV}))for t > 0, r > 0 and a hitting time r.

The next result (a generalized multivariate version of the result in

Willekens [40]) will be of relevance for the subsequent study of func¬

tionals of additive processes, but is also interesting in itself. It says es¬

sentially the following. If the tails of the process are sufficiently heavy,

then the probability that the process reaches a set far away from the

origin before time t > 0 is asymptotically equal to the probability that

the process ends up in that set at time t. Note that condition (2.11) is

much weaker than that of multivariate regular variation.

For positive r, u and T let

arjT(u) = sup{PSjt(x, B^r) :xGRd and s,t E [0,T],t- s G [0,u]}.

Theorem 2.7 Let (Xt)t>o be a strong Markov process on R and let

A G Vß be bounded away from 0. For a sequence (an) satisfying 0 <

an f oo let rn = infjs : Xs G anA}. Fix t > 0, let r > 0 be arbitrary

and put Sr(anA) = {x G Md : infyGanA |x — y| < r}. Then

(i) F(Xt G Sr(anA)) > F(rn <t)(l- ar,t(t))

(ii) If arjt (t) —y 0 as r —> oo and for any r > 0

P(X g Sr(ar,A))=

(2u)

n^oo p(Xt G anA)' v '

then

lim J^L- = i.n^oo P(Xi G anA)

Proof. Write t = rn. Clearly,

P(r < t) = F(t <t,Xte Sr(zA)) +P(r < t,Xt G Sr(zA)c)

< F(Xt G Sr(zA)) + P(r <t,Xte Sr(zA)c)

< F(Xt G Sr(zA)) + P(r < t, Xt - XT G ßg>r).

By the strong Markov property,

F(r<t,Xt-XTeBc0jr) = E(l{T<nl{Xt_xTGBoV})= E(l{T<nE^x^(l{Xt_xTGi?oV}))< E(l{T</}ar,t(t))= F(T<t)arit(t).

Hence, P(Xt G Sr(zA)) > P(r <t)(l- ar:t(t)). Furthermore,

, . r•

tnrn<t)

^vF(rn<t)

1 < hm ml —— jt-< hm sup

nôo P(Xf G anA)~

nôo F(Xt G anA)

F(Xt G Sr(anA)) 1< lim sup

nôo P(Xt G anA) 1 - ar,t(t)1

1 - arjt(t)'

Since r > 0 was arbitrary we can let r —)• oo from which the conclusion

follows. D

Before we proceed we note that for additive processes the condition

®r,t(t) —» 0 as r —)• oo always holds.

Lemma 2.8 Let (Xi)i>0 be an additive process on Mr. Then, for every

t > 0, arj(t) —y 0 as r —>• oo.

Proof. By Theorem 10.4 p. 55 in Sato [38], (Xf)t>0 is a Markov process

with spatially homogeneous transition function

PU;V(x, B) = F(XV - Xu G B - x), 0 r)^0u,v£[0,t]

as r —y oo. D

Theorem 2.9 Le£ (Xi)i>0 èe a separable strong Markov process on

Fix an arbitrary t > 0 and suppose that ar^(t) -> 0 as r —>• oo. XTien

£/ie following statements are equivalent.

nF(a~1Xt e-)A pt(-) on B(Md\{0}).

(ii) There exist a sequence (an), 0 < an f oo, and a nonzero Radon


nPfû^X, G • some s < t) A pt(-) on B(Md\{0}).

The sequences (an) and the measures pt in (i) and (ii) can be taken to

be equal.

Proof, (i) =>• (ii) Let Vu,s = {x G Rd\{0} : |x| > u,x/|x| G S} for

u > 0 and let S G #(8d_1) be a /it-continuity set. Suppose first that

ßtiYu,s) = 0. For every 8 G (0,1) there exists nrj such that for n > nrj$

Sr(anVu,s) C an(l - ô)VUjs U an(l - S)Vu,Ss(s)\s,

where Sr{anVUjs) = {x G Rd : MyeanVuS |x - y| < r} and SÔ(S) =

{x G Sd~1 : infy<Es |x — y| < ô}. Furthermore, VUjss(s)\s fails to be a

^-continuity set for at most countably many 6. Hence, by Theorem 2.7

(i), for n > nrjS,

nF(a~1Xs G VU:s some s < t)

<nF(a-1Xt G (l-ô)Vu,s)

+nP(a"1Xt G (1 - <5)KÄ(s)\s)

^0 +

1 — «r,« (*) l-"r,<W

(i-(y)-a^(K,5J(5)\g)1 -ar7t(t)

for some a > 0. Since <5 G (0,1) was arbitrary we can let ô —y 0 and the

conclusion follows.

Now, suppose pt(VUjs) > 0. We first show that the condition of The¬

orem 2.7 (ii) is satisfied. Take r > 0. For every ô G (0,1) there exists

nrjs such that for n > nTj$

Sr(anVu,s) C an(l - ô)VUjs U an(l - 5)VUjSg(s)\s-

Hence

1 <P(Xt G Sr(anVu,s))

F(Xt G anVu,s)

<P(Xt G aw(l - ô)Vu,s) P(Xt G flw(l - £)K^(s)\s)

P(X* G anVu,s) F(Xt G anK,s)

_^pt((l - ô)VU:s)

|Mt((l-^)K,gg(g)\g)

MK,s) /J*(K,s)

Vt(VUiss(s)\s)\= (l-S)-a 1 +

Mt(K,s)

for some ck > 0 as n —>• oo. Since ô G (0,1) was arbitrary, by letting

<5 —^ 0 it follows that the condition in Theorem 2.7 is satisfied. Hence,

by Theorem 2.7,

P(Xg G anVu,s some s <t)_

n^So P(Xt G anVUts)

and thus

nP(Xs G anVUjs some s < t)

wy r- \r ^vX* G awK,g some g <t)=

nF(Xt G anVu,s) F,Y c „ t/ ^y M*(K,s),

as n —> oo. Since convergence of every such set VUjs implies vague

convergence on B(M \{0}) the conclusion follows.

(ii) => (i) For any A G ß(Rd\{0}) we have F(a~1Xt G A) < F(a~1Xs G

A some s <t) and hence

limsupnP(a^1Xt G A) < pt(A).n—)-oo

A lower bound is constructed as follows. Let VUjs = {x G Rd\{0} :

|x| > u, x/|X| G S} for u > 0 and 5 of the form {x G §d_1 : |x - x0| <

ro} for some xq G Sd~1 and ro > 0, such that VU}s is a /^-continuity

set. For such an S and small enough ô > 0, let S6(S) = {x G 5 :

infyGSd-i\s |x - y| > 6} and for r > 0 let Sr(anVU:S) = {x G a„K,s :

infyGanyc |x — y| > r}. For every ô G (0,1) there exists nrj such that

for n > nrj,

Sr(anVUjS) D a„(l + £)K,s«(s)-

Hence, for n > nr^,

nF(a~1Xt e Vu,s)

> nF(a~1Xs G (1 + ô)VU}S6(S) some s <t,

Xq - Xs G anB07s all q G [s, t])

> nF(a~1Xs G (l + S)VUisHs) some s < t)(l - a^/^) \

-^(l + ô)-apt(Vu,sS(s)),

by combining the strong Markov property, the assumption arj(t) —y 0

as r —y oo and Lemma 2 p. 420 in Gihman and Skorohod [20]. Since

ô > 0 was arbitrary we can let ô —y 0 and hence

liminfnP(a-1Xf G Vu,s) > Pt(Vu,s).

Since convergence of every such set VU}s implies vague convergence on

ß(Rd\{0}) the conclusion follows. D

We now turn our attention to vectors of functionals applied to each

component of a multivariate additive process. We consider the implica¬

tions of regular variation of the process at time t on the vector of the

componentwise suprema of the process and the componentwise suprema

of its jumps up to time t. We have the following result.

Theorem 2.10 Let (Xt)t>o be an additive process on Md with system

of generating triplets ((At, i^,7f))t>o. Fix an arbitrary t > 0. Then the




nF(a~1Xt e-)A pt(-) on B(Md\{0}).

(ii) There exist a sequence (an), 0 < an f oo, and a nonzero Radon


nPfû^X, G • some s < t) A pt(-) on B(Md\{0}),

The sequences (an) and the measures pt in (i) and (ii) can be taken to

be equal. Moreover, if any of the statements (i) or (ii) hold, then

(iii) nP(a"%A G • ) A pt(-) on B(Md+\{0}),where XA = (sup0<s (ii) follows from Theorem 2.9.

(i) => (iii) Since (Xs)s>0 is an additive process there exists Q0 G T

with P(^o) = 1 such that, for every u G Qo-, Xs(u;) is right-continuous

in s > 0 and has left limits in s > 0. Hence XA > 0 for every wgOq

and s > 0. Let £ be the Poisson random measure given by (2.5). Then,

for any xGl}\{0},

P(Xf < x) = P(£((0,i] x [-oo,x)c) = 0) = exp(-^([-oo,x)c)).

Hence

P(XA < x) = { exP(-^([-°°>x)C)) fOT x e M+\{°}510 otherwise.

By Theorem 1.8, pt(d[—oo,x)c) = 0 for every x G R+\{0} (boundariesof spheres centered at 0 have zero /zt-measure). Hence, by Theorem 2.3,

for every x G R+\{0}

nP(a-%A G[-oo,x)c)

= n(l - exp(-z/t(an[-oo,x)c)))

= n^(an[-oo,x)c)(l + 0(^(an[-oo,x)c)))

^/it([-oo,x)c).

Take a, b G R^_\{0} with a < b. Then

nP(an-%AG[a,b))

= E (-^)il+-+id+1nF(a-1X^ G {[-oc,xltl) x ... x [-oo,^J}c),ü,-..)*d{l,2}

where Xj\ = a^ and Xj2 = b^ for every j G {1,..., d}. Hence

nPK^XfG^b))^^^)).

Since convergence of every such set [a, b) implies vague convergence on

#(R+\{0}) the conclusion follows.

(i) => (iv) Take x G R+\{0}. Note that if we would only take x G

(0, oo)d then we would not end up with a convergence determining class.

Since X^ > Xt a.s. and pt(d[x., oo)) = 0 (boundaries of spheres centered

at 0 have zero /it-measure)

nPK^X* G [x,oo)) > nF(a~1Xt G [x, oo)) ^ ^([x, oo)),

i.e. liminfj^oo nP(a~1X|c G [x, oo)) > ^([x, oo)). To complete the

proof it remains to show that limsupn_).00 nP(a~1X^ G [x, oo)) <

Pt([x, oo)). First, define

A(1_e)x = {zeMd:z^>(l-e)x^,j = l,...,d}

C%} = {z G Md : z G [-oo, x)c, z^ G [-oo, (1 - e)x^)}

Dgl = {z G Md : z G [-oo, x)c n Ac(1_e)x1 z G [x(fc), oo]}

and note that for each k G {1,..., d}, [—oo, x)c C A(i_e)x U C^J U Dxjand that the sets on the right-hand side are disjoint. If X^ G Ax,

then either Xs G A(!_e)x for some s < t or, for some A; G {1,..., d],

XSl G Cx,e for some s\ < t and XS2 G -Dx,i for some s2 <t,S2=/z s\.

Assume without loss of generality that

P(Xai G C^x,e,XS2 G Da%e some Sl,s2 < t)

> F(XS1 G CikXe, XS2 G £><£>x>e «orne Sl,s2< t)

for fc = 2,..., d. Then

nP(X* Gan[x,oo))

< nP(Xs G Aan(1_e)x some s<t)

+nP(ug=1{Xfll G C£Xj£>X8a G Df)X]£ some 5l,s2 < t})

< nP(Xs G Aan(i_e)x some s < t)

+ndF(XSl G C£>x>eîX82 G L»^ some 5l)52 < t)

By Theorem 2.7

nP(Xs G Aan(i_e)x some s<t)

~nF(Xt GAan(1_e)x)-^ Mt((l - e)[x, oo)) = (1 - e)"a/ii([x, oo)),

as n —y oo. Let 7 = inf cr,(i) cn(i) |u — v|. Then 7 > 0 and

ndP(XSl G C£Xf£,Xfl2 G Z^ some 5l,s2 < t)

< ndP(XSl G Ci;)X]£,XS2 G D^ some 5l < s2 < t)

+ ndF(XSl G C^, XS2 G Da%je some s2 < 5l < t)

< ndF(XSl G Ci;)X]£,XS2 - XS1 G 5J>an7/2 some 8l < s2 < t)

+ ndF(XSl G D^x>e, XS2 - XS1 G ßS>0n7/2 some 8l < s2 < t)

< ndF(Xs G C<£Xte some s < t) F(XS G Bc0^ß some s < t)

+ n<2P(Xs G 2#Jx>e some s < t) F(XS G ßS,an7/4 some * < *)

—>• 0 as n —> 00.

Since e > 0 was arbitrary it follows that limsupn_).00nP(a~1X£ G

[x, 00)) < pt([x., 00)). Hence

nF(a~1X; G [x,oo)) ^/if([x,oo)).

Take a, b G R+\{0} with a < b. Then

nP(a-1X;G[a>b))

= J2 (-l)il+-+idnF(a-lXl G [xiil5oo) x ••• x fed,oo)),H,...,id6{l,2}

where £ji = a^^ and #j2 = &(" for every j G {1,..., d}. Hence

fiP(a;1Xt*G[a)b))^/it([a,b)).

Since convergence of every such set [a, b) implies vague convergence on

#(R^_\{0}) the conclusion follows. D

Assertion (ii) of the Theorem 2.10 gives the asymptotics for the prob¬

ability that the process reaches a certain set during the time inter¬

val [0,t]. What if we would allow the set of interest to vary over

time? I.e. we are looking for the asymptotics of the probability that

the graph of the process, {(s,Xs) : 0 < s < t], intersects a rela¬

tively compact set in #([0,£] x (R \{0})), the cr-algebra generated by

the sets of the form T x B with T G B([0,t]) and B G #(1^(0}).This requires knowledge of the tail of Xs for (almost) all s G [0,t],not only of Xt as has been the case so far. To obtain this kind of

result we will work with the measure v (see Theorem 2.1) instead of

simply ut. In order to be able to use the vague convergence frame¬

work we extend the measure v to [0, oo) x (R \{0}) by requiring that

u({(s,x) : s = 0 or x G R \Rd}) = 0. This extension is unique. Let us

introduce the operation * such that for a G (0, oo) and sets of the form

A x B, A G B([0,t]) and B G #(Rd\{0}), we have a * A x B = AxaB.

Clearly this operation can be extended to all sets in B([0, i]xl \{0}).

Theorem 2.11 Let (Xt)t>o be an additive process on Md with system

of generating triplets ((At, vt,^t))t>o- Fix an arbitrary t > 0. Then the


(i) There exist T C [0,t] such that 0,t G T and such that [0,t]\T is

at most countable, a sequence (an), 0 < an f oo, and a collection

of Radon measures {ps : s £T} on B(M \{0}) such that for every

s G T, ps(M \Md) = 0, pt is nonzero, and

nF(a~1Xs G • ) A ps(-) on B(m\{0}). (2.12)

(ii) There exist T C [0,t] such that 0,t G T and such that [0,t]\T is

at most countable, a sequence (an), 0 < an f oo, and a collection

of Radon measures {ps : s G T} on B(M \{0}) such that for every

s G T, ps(M \Md) = 0, pt is nonzero, and

nvs(an-)^ps(-) onB(Md\{0}). (2.13)

(iii) There exist a sequence (an), 0 < an f oo, and a nonzero Radon

measure p on B([0,t] x (Rd\{0})) with p([0,t] x Rd\Rd) = 0 such

that

nV(an * • ) A p(-) on B([0, t] x (Rd\{0})). (2.14)

(iv) There exist a sequence (an), 0 < an f oo, and a nonzero Radon

measure p on B([0,t] x (Rd\{0})) with p([0,t] x Rd\Rd) = 0 such

that

nP({(«, Xs) : 0 < s < t} n (an * • ) ^ 0) A /!(•) (2.15)

on B([0,t]x (Rd\{0})).

Furthermore, the sequences (an) in (i)-(iv) can be taken to be equal and

then the measures ps and p([0, s]x ) in (i)-(iv) coincide on B(M \{0})

for every s G [0,i\.

Remark 2.12 As seen from the proof below, (iv) =>• (i) with T = [0,t]and (iii) =>• (ii) with T = [0,t]. Hence we can without loss of generality

let T = [0,t].

Definition 2.13 A stochastic process on R satisfying Theorem 2.11

(iv) is said to have a regularly varying graph on [0,t].

Note that v({s] x B) = 0 for every s G [0,t] and B G #^^{0}).However, the limit measure p may charge sets of the form {s} x B as

seen in the following example.

Example 2.14 Let v be a probability measure on [0,1] x [1, oo) given

by

oo

v(dt x dx) =^2kl^i/2-i/(2k),i/2+i/(2k))(t)dtl[kik+i)(x)ax~a~1dx.fc=i

Let £ be a Poisson random measure with intensity measure v, and let

X = (Xt)tç\o,\\ be a stochastic process given by

Xt = / x£(ds x da:).J[0,t]x[l,oo)

For any t G (0,1),

ù([t - l/(2m),t + l/(2m)] x [1, oo))

< £([1/2 - l/(2m), 1/2 + l/(2m)] x [1, oo))oo

= J2 (k~a -(k+1)_a) = m~a -* °

k=m

as m —y oo. Similarly, v([0, l/m] x [1, oo)) —y 0 as m —y oo and v([l —

l/m, 1] x [1, oo)) —y 0 as m —y oo. Hence, for s < t, v([s, t] x [1, oo)) —)• 0

as \t — s\ —y 0. Thus, for any e > 0,

F(\Xt-Xs\>e) < F(£([s,t]x[l,oo))>0)

= 1 - exp{-v([s,t] x [l,oo))| -> 0

as |t — s| —y 0, i.e. X is stochastically continuous. Moreover, by Propo¬

sition 19.5 p. 123 in Sato [38], for disjoint T1:... ,Tk G B([0,1]),

/ x£(ds x da;),..., / xÇ(ds x da:)JTix[l,oo) JTfex[l,oo)

are independent, i.e. X has independent increment. Moreover, by con¬

struction, X(uS) is right-continuous with left limits for every u G Q,.

Finally, Xq = 0 a.s. Hence, X is an additive process. By Proposi¬

tion 19.5 p. 123 in Sato [38], for every t G [0,1], Xt has Levy measure

ut(-) = £([0,t] x • ). Take u > 0. For t > 1/2 and n large enough,

/>oo

rw([0, t] x (n1//aw, oo)) = n ax~a~1dx = u~a.Jn1/au

For t = 1/2,

1 1°°nù([0,t] x (n1/au, oo)) = n- j

For t < 1/2 and n large enough,

n£([0,t] x (n1/au,oo)) = 0.

Hence, for every t G [0,1],

nvt(nl'a -)^pt(-) on£((0,oo]),

Iax~a~1dx = -u~a.

where

T 0 if t< 1/2,tH(B)=i \ JB ax~a~ldx \ît =1/2,

{ JBax~a-1dx ift> 1/2.

In particular, p({l/2} x B) = ^ JBax~a~1dx which clearly may be

nonzero.

The following Lemma is needed for the proof of Theorem 2.11.

Lemma 2.15 Let (Xt)t>o be a strong Markov process on R .Fix an

arbitrary t > 0 and suppose that cvrj(t) —>• 0 as r —y oo. Then, for any

sequence (an) with 0 < an t °°;

nF(a-1Xt G • )-H>0 onB(Md\{0}) (2.16)

z/ and only if

nF(a~1Xs G • some s<t)^0 on B(Md\{0}). (2.17)

Proof. Take a sequence (an), t > 0 and a relatively compact B G

ßor'uo}).Suppose that (2.17) holds. Since

P(X* G anB) < F(XS G anB some s < t),

(2.16) follows immediately.

Suppose that (2.16) holds and, without loss of generality, that B cMd.

Fix r > 0 and let 7 = infxGs |x|. By Theorem 2.7 (i),

Xs G anß some s <t) < F(XS G ünBg^ some s <t)

<[Xt G fl.(qwflS,7))

l-ttr;f(t)

For every r > 0 there exists an nrj7 such that Sr(anBg^) C an-Bo ,2

for n > nri7. Hence

nF(XteanBc /2)limsupnP(Xs E anB some s <t) < lim sup

'—

n—)-oo n—>-oo -L C^r,i i^J

= 0,

from which (2.16) follows. D

Proof of Theorem 2.11. (i) <^> (ii) For every s G T for which ps

is nonzero this equivalence has been established in Theorem 2.3. We

need to establish the equivalence also if ps = 0. Since pt is nonzero,

by Remark 1.13, the sequence (an) has to be regularly varying. Hence

equivalence can be proved by exactly the same arguments as in the proofof Theorem 2.3.

(ii) =>- (iii) Define the Radon measure p on #([0,£] x (R \{0})) by

p([0,s] x B) = ps(B) for every s G T and B G BQÜ^O}). Since such

sets of the form [0, s] x B form a 7r-system which generates #([0,£] x

(R \{0})) this uniquely defines p. We first need to show that the set

{nv(an * • )} is relatively compact in the vague topology. For each

bounded set A G #([0,£] x R \{0}) there exists a bounded set B G

#(Rd\{0}) such that A c [0,t] x B. Hence,

supnu(an * A) < supnvt(anB) < oo,n n

from which it follows by Theorem 1.4 that {nu(an * • )} is vaguely

relatively compact. Let p! be a vague limit of the subsequence (n'v(an> *

• )). Fix an arbitrary s G T and relatively compact B G B(M \{0}) with

pt(dB) = 0. Then ps(dB) = p([0, s] x dB) < p([Q, t] x dB) = pt(dB) =

0 and hence

n'v(ani * [0,s] x B) = n'us(an'B)

-+ ps(B) = p([0,s]xB),

i.e. /TQOjS] x B) = /l([0,s] x B). Since the sets of the form [0,s] x B

with s ET and B G 23(R \{0}) with pt(dB) = 0 form a 7r-system which

generates #([0,£] x (R \{0})) the conclusion follows.

(iii) =>• (ii) Put ps(-) = p([0,s] x • ) for every s G [0,t]. Let F G

d

B(M \{0}) be closed and bounded. Then [0, s] xF is closed and bounded

and by Theorem 1.1 and Remark 1.2,

lim supnvs(anF) = lim supnv(an * [0, s] x F)n—)-oo n^-oo

< p([0,s]xF) = ps(F).

d

Hence, by Theorem 1.1 and Remark 1.2, nus(an • ) -^ ps(-) on B(M \{0})for every s G [0,t].

(i) =>• (iv) Define the Radon measure p, on #([0,£] x (R \{0})) by

p([0,s] x B) = ps(B) for every s G T and B G #(^{0}). Since such

sets of the form [0, s] x B form a 7r-system which generates #([0,£] x

(R \{0})) this uniquely defines p. We first need to show that the set

{nP({(s,Xs) : s G [0,t]} D (an * • ) ^ 0)} is relatively compact in

the vague topology. For each bounded set A G #([0,£] x R \{0}) there

exists a bounded set B G B(M \{0}) such that A C [0,t] x B. Hence,

by Theorem 2.10,

supnP({(s,Xs) :s£ [0,t]} n (an * A) ^ 0)n

< supnP(a~1Xs G B some s < t) < oo,n

from which it follows by Theorem 1.4 that {nP({(s,Xs) : s G [0,t]} n

(fln * • ) 7^ 0)} is vaguely relatively compact. Let p' be a vague limit

of the subsequence (n'P({(s,Xs) : s G [0,t]} n (an' * • ) ^ 0)). Fix an

arbitrary s G T and relatively compact 5 G #(R \{0}) with pt(dB) =

0. Then, by Theorem 2.10 and Lemma 2.15,

n'F({(s,Xs) : s G [0,t]} n (on* * [0,s] x5)^)

= n' P«/Xu G 5 some u < s) -> //S(S) = /I([0, s] x 5),

i.e. /i'([0,s] x B) = /l([0,s] x S). Since the sets of the form [0,s] x 5

with s eT and ß G B(M \{0}) with pt(dB) = 0 form a 7r-system which

generates #([0,£] x (R \{0})) the conclusion follows.

(iv) => (i) Put ps(-) = p([0,s] x • ) for s G [0,t]. Let F G #(Rd\{0})be closed and bounded. Then [0, s] x F is closed and bounded and by

Theorem 1.1 and Remark 1.2,

limsupnP(a~1Xw G F some u < s)n—>-oo

= limsupnP({(s,Xs) : s G [0,t]}n{an*[0,s] x F) ^ 0)n—>-oo

<M0,s]xF)=|is(F).

Hence, by Theorem 1.1 and Remark 1.2, nP(a~1Xu G • some u < s) A-

Ps(-) on B(M \{0}) for every s G [0,t]. Combining Theorem 2.10 and

Lemma 2.15 now yields nF(a~1Xs G ) A ps(-) on #(Rd\{0}) for

every s G [0, t]. D

Working on the product space [0, t] x (M \{0}) allows us to consider

further interesting functionals such as the integral of each component

of the process. To derive the tail behavior for this functional we use

the intuitive argument described in the introduction which says that if

the process takes a big jump, then it varies very little before and after

the jump, compared to the size of the jump. Intuition then tells us that

the integral of component j, say, becomes bigger than u if Xg exceeds

u/(t — s) for some s G [0,t]. That is, the integral of X^ becomes

bigger than u if the graph {(s,Xg ) : 0 < s < t} intersects with the

set a[3) = {(s,x) G [0,t] x (R\{0}) : x > u/(t - s)}. This argument is

made precise in the final result of the chapter.

Theorem 2.16 Let (Xt)t>o be an additive process on Md with system of

generating triplets ((At,vtilt))t>o- Fix an arbitrary t > 0 and suppose

there exist a sequence (an), 0 < an t °°? and a nonzero Radon measure

p on B([0,t] x (Rd\{0})) with p([0,t] x Rd\Rd) = 0 such that

nu(an * • ) A p(-) on B([0, t] x (Rd\{0})).

-rrtv(i)A„ rt v(d)// It = (f0 X^ds, ...,j; Xsa)ds), then

nF(a-% e-)A £({(*,x) G [0,*] x (r"\{0}) : x G-!-

• })t — s

on B(M \{0}). In particular, if (Xs)s>0 is a Levy process, then

nF(a-1lt G • ) A ^rMO on B(m\{0}),a + 1

where p(-) = p([0,1] x • ) and a > 0 is such that p(u • ) = u~ap(-) for

all u > 0.

Proof. By the Lévy-Itô decomposition we may write the process (Xs)as the sum of three independent processes. With the notation of Theo-

rem 2.1,

x» = xJH

+ lim/ {x£(d(it, x),a;) — xi/(d(it, x))}e±° J(o,s]xDeA

+ / xf(d(u,x),o;),J(0,s]xDliOO

J.,

on Qi, where (X^) has a version with continuous sample paths, (Ys) is

a jump process with small jumps, and (Js) is a jump process with big

jumps. For j = l,...,m, let Vf = Jtj/m - Jt(j-i)/m- For u G fi0nßi,

s as m -^ oo,

and

5^|V^(o;)|-)> 5Z lAJ«MI asm^oo,

J= l u(0,<)

where the sum extends over all (finitely many) u such that |AJu(a;)|is nonzero. Take e G (0,1/2) and let A be a relatively compact /it-

continuity set of the form {x G Rd\{0} : |x| > w,x/|x| G S} for

u > 0 and S G jB(8d_1). Convergence of every such set implies vague—d

convergence on B(M \{0}). To begin with, we study the behavior of

nP(J0 Jsds G anA) as n —y oo. Let 7 = infxfE,4 |x| and set At = {(s, x) :

x G t^A, s G [0, t]} G #([0, t] x (Rd\{0})). We begin by constructing a

lower bound.

m,

n¥C>Z—Jtj/m eanA)3= 1

>nP(U=1{V G \ flw(l + e)A,^|Vri<ttne7/t})t — ti/m

,

m1

= EF(E lV"l < *nej/t) nF(Vf G flw(l + e)A)

771 771 -.

>P(ElVn <<WA)£«P(V -——a„(l + e)^)j=l 3=1

•"

7711

> F(E lV7l ^ «W*) «P(U7=i{V7 G flw(l + e)A}).r—' J J t — ti rn

3 = 1'

Letting ra —>• oo, we arrive at

nF(J Jsds e anA) > F( ^ |AJ„| < anej/t) nv((l + e)an * At).

We now construct an upper bound.

771,

nF£-J^/-Ga-A)lit

3 = 1

< n¥(Uf=1{VT e t-r/m^1 " e)A' ^ |Vrl"

°nC7/*})J/m

i^3

+ nF(Jti/meB^anei/t,

Jtj/m ~ Jti/m G 50,an67/t SOme « < j < "l)

1

£ — £j/ra

+ «P(J«/mGSoian£7/t>

Jti/m - Jti/m G ß0,ane7/* SOme * < 3 < )

1

t—

tj/m

+ nF(Jtj/m G £o,a„e7/t some 3 < )2-

< nP(Uf=1{V7 G ——^-^(l - e)A})

< nP(Uf=1{V7 G^

,,^an(l - e)A})

Letting m —y oo, we arrive at

et

nP( / Jsds G anA)Jo

<rw((l-e)an*At)+nF(Js G £o,ane7/* somes G (0,t])2.

Since limn_),ooP(^uG/0iN |AJW| < ane7/£) = 1 and by the second part

of Theorem 2.7 lim-^ nP(Js G -B^ a ,tsome s G (0, £])2 = 0, we get

liminf nP( / Jsds G ani) > p((l + e) * At),n^°° Jo

limsupnP(/ Jsds G ani) < p((l - e) * At).n—>-oo Jo

Since e > 0 was arbitrary and p(u*- ) = u~ap(-) for all u > 0, it follows

that the lower and upper bound coincide, i.e.

lim nP( / Jsds G anA) = p(At).7WOO Jo

Since as n —)> oo

r 1

nP( / Xjds G an4) < nP(Xj G -anA some s < t) -> 0

Jo ^

and

/** i

nP( / Ysds G anA) < nF(Ys G -anA some s < t) -y 0,jo ^

by Theorem 1.28,

lim nP( / Xsds G anA) = lim nP( / Jsds G anA).n-»-oo J0 n->-oo JQ

Hence

lim nP( / Xsds G anA) = //(At).n^°° Jo

In the case of a Levy process, i.e. if /x([0,s] x • ) = sp(-) for s G [0,i\.then

KAt) = [ P(T^-A)ds = p(A) [ (t - sTds = ^-\p(A).Jo t - s Jo a + !

D

Chapter 3

Regular variation for

general stochastic

processes

In this chapter we consider a different and more general approach to

heavy tail analysis for stochastic processes. We will however make sev¬

eral comparisons with the results derived in Chapter 2. Multivariate

regular variation which was discussed in detail in Chapter 1 provides

a natural way for understanding the tail behavior of heavy-tailed ran¬

dom vectors. The definitions of regular variation for random vectors

can be modified so that the modified formulations make sense for more

general state spaces; similar formulations are possible for stochastic pro¬

cesses with sample paths in D([0, l],IRd), the space of Revalued right-

continuous functions on [0,1] with left limits. These formulations seem

to be well suited for understanding the tail behavior of heavy-tailed

stochastic processes. We will exemplify this in various forms through¬

out the chapter. Recall that a random vector X on Md is said to be

regularly varying if there exist an a > 0 and a probability measure a on

the unit sphere Sd~l = {x G Md : |x| = 1} such that, for every x > 0,

65

66 Chapter 3. Regular variation for stochastic processes

as u —y oo,

P(|X|>us,X/|X|e.),x-„j() „.»(s-i-ix

where B(Sd_1) denotes the Borel cr-algebra on Sd~1 and -^ denotes weak

convergence. The probability measure a is referred to as the spectral

measure of X. It describes in which directions we are likely to find

extreme realizations of X. Similarly, we say that a stochastic process

X = (Xt)iG[oti] with sample paths in Z)([0, l],K.d) is regularly varyingif there exist ana>0 and a probability measure a on £>i([0,1], Md) =

{x G D([0,l],Md) : supfr0 ]_] |xi| = 1} such that, for every x > 0, as

u —y oo,

P( X oo> UX,X/\X\00 G • ) _

, ,

o/T-i /rn il in>d\\

p(|xu>u)>* "(•) onB(Dl([0,l],M)),

where B(D1([0,l],Md)) denotes the Borel cr-algebra on £>i([0, l],Md)and Ixloo = supfGr0 ^ \n.t\- The spectral measure a contains essentially

all relevant information for understanding the extremal behavior of the

process X. For example, it might be of interest to know under which

conditions the extremes of X are due to (at most) one single extreme

jump (we allow also an extreme starting point). This can be formalized

in terms of the support of the spectral measure by showing that the

spectral measure concentrates on step functions, i.e. on the set

{xGJDi([0,l],Rd):x = ylK1],^G[0,l],yG§d-1}.

We show that this is the case for a large class of regularly varying Markov

processes, including all regularly varying additive processes (and hence

also all regularly varying Levy processes).

A natural question is why one would prefer formulating regular variation

on D([0, l],Ed) rather than on, say, (E^)^0'1]. The main reason is that

many interesting mappings from D([0,1], Md) to D([0,1], Md) (or to Mk)are continuous whereas the corresponding mappings from (R^)^0'1] are

not. However, with constructions similar to the one used in this chapter,

regular variation can be formulated on other complete separable metric

spaces. In this chapter we prefer to work on D([0,1], Md). An equivalent

67

definition of regular variation on D([0,1], Md) is the following; a stochas¬

tic process X with sample paths in D([0, l],Kd) is regularly varying if

there exist a sequence (an), 0 < an t oo, and a nonzero boundedly finite

measure m on B(D([0,l],Md)) with m(D([0, l],Rd)\D([0, l],Md)) = 0

such that, as n —y oo,

nP(a"1XG •) Âm(-) onB(D([0,l],Md)), (3.1)

where A denotes so-called ^-convergence. (The precise meaning of

D([0, l],Md) is explained in Section 3.1. At this point it may be viewed

as only a slight modification of D([0,1], IRd) needed in order to use the

concept of Û7-convergence.) Let h be a positively homogeneous (i.e.

fo(Ax) = A/i(x) for A > 0) measurable mapping from D([0, l],IRd) to

D([0,1], M?) (or to Mk). Then, if (3.1) holds and if h satisfies some mild

conditions, as n —y oo,

nF(a~lh(X) G • ) A m o h~1{ • n D([0,1], Md)) on B(D([0,1], Md))(3.2)

k

(or on B(M \{0}), M = [—00,00]), i.e. we have a version of the Con¬

tinuous Mapping Theorem. Hence, under mild conditions on h, regular

variation of X implies regular variation of h(X) and we can express its

limit measure in terms of m and h as in (3.2). In Section 3.1 we state

the two definitions of regular variation on D([0, l],Md) and show that

they are equivalent. Moreover, we give necessary and sufficient condi¬

tions for regular variation for a general stochastic process with sample

paths in D([0, l],IRd). Finally, we give a continuous mapping theorem

which provides a powerful tool in the subsequent analysis. In Section 3.2

we focus on strong Markov processes with asymptotically independent

increments (see Section 3.2 for the precise meaning of asymptotically

independent increments). We obtain sufficient conditions for regular

variation for such processes which are easier to verify since they involve

only the marginals X^ of the process X. Moreover, we show that the

limit measure m of such regularly varying Markov processes vanishes

on Ve where

V = {x G £>([0, l],Ed) : x = yl[W)1],t; G [0, l],y G Md\{0}}.

This means that, asymptotically, the process reaches a set far away from

the origin either by starting there or by making exactly one big jump to


this set and, in comparison to the size of the jump, it stays essentially

constant before and after the jump. For an illustration, see Figure 3.1

which shows eight simulations of

x | {\xs\ > i^ixV0-9)for some s G I°> X]}'

where X is a Levy process with X\ having a Cauchy distribution with

density fxx(x) = l/(ir(l + x2)). On one hand this means that we are

able to quantify the idea of one big jump in terms of the support of

the regular variation limit measure. On the other hand, and equally

important, this in combination with the Continuous Mapping Theorem

(3.2) allow us to explicitly compute tail probabilities of h(X) for many

interesting choices of h. See e.g. Examples 3.20 and 3.21 with

Mx) = (vte[o,i] *e[o,i]

(1) (d)sup xt ,..., sup xt

and

h(x)=(J1x[1)dt,...,J1xld)dt)respectively. In Section 3.3 we study filtered stochastic processes of the

form

Yt= f f(t,s)dXs, te [0,1], (3.3)Jo

where X is a regularly varying Levy process (i.e. a strong Markov pro¬

cess of the type studied in Section 3.2) with sample paths of finite

variation. Under the assumption that the kernel / is continuous we

show that Y can be viewed as a mapping of the process X, which is

sufficiently regular to satisfy the conditions of the Continuous Mapping

Theorem. We show that Y is regularly varying and determine the limit

measure.

In order to make the presentation as readable as possible and in order to

focus the attention on the underlying ideas rather than on technicalities,

we give the proofs at the end of each section.

3.1. Regular variation on D 69

3.1 Regular variation on D

Let us introduce regular variation on D = D([0, l],Rd); the space of

functions x : [0,1] —y Md which are right-continuous with left limits.

This space is equipped with the so-called Ji-metric (referred to as do in

Billingsley [5]) which makes it complete and separable. The formulation

of regular variation we will use has recently been introduced in de Haan

and Lin [14] in connection with max-infinitely divisible distributions on

D. See also Giné, Hahn and Vatan [18].

We denote by D\ = £>i([0,1], Md) the subspace {x G D : suptGr01i |xt| =

1} equipped with the subspace topology. Define D = (0, oo] x _Dl5 where

(0,oo] is equipped with the metric p(x,y) = \l/x — l/y\ making it

complete and separable. Then D is a complete separable metric space.

Note that to each nonzero function x G D corresponds a unique element

(#*,x) G D where x* = supfGr01-i |x^| and x = x/x*. For x G D we

write Ixloo = suptGr0-n \x.t\ and for x = (x*, x) G D we write jx^ = x*.

A consequence of the above construction is that

B(D)n(D\{0}) = B(D)n(D\{0}),

i.e. the Borel sets we are interested in are the usual Borel sets on D

which do not contain the zero function.

We will see that regular variation on D is naturally expressed in terms

of so-called w-convergence of boundedly finite measures on D, i.e. mea¬

sures which assign finite measure to bounded sets. A sequence of bound¬

edly finite measures (mn)n<=^ on a complete and separable metric space

E converges to m in the tû-topology, mn A- m, umn(B) —y m(B) for ev¬

ery bounded Borel set B with m(dB) = 0. If the state space E is locally

compact, which D is not but M \{0} is, then a boundedly finite mea¬

sure is called a Radon measure, and w)-convergence coincides with vague

convergence and we write mn -^ m. Finally we note that if mn A m

and mn(E) —>• m(E) < oo, then mn A m. For details on w-, vague- and

weak convergence we refer to Appendix 2 in Daley and Vere-Jones [12].See also Kallenberg [26] for details on vague convergence.

Definition 3.1 A stochastic process X = (Xt)t^[o,i] with sample paths

in D is said to be regularly varying if there exist a sequence (an),0 < an t °°; and a nonzero boundedly finite measure m on B(D) with

m(D\D) = 0 such that, as n —y oo,

nF(a~1X e ) A m(-) onB(D). (3.4)

If Definition 3.1 holds, then the limit measure m has the following scal¬

ing property (the proof is identical to that of Theorem 1.14, with the

obvious notational changes, and therefore left out).

Theorem 3.2 The limit measure m has a scaling property; there exists

an a > 0 such that m(uB) = u~am(B) for every u > 0 and B G B(D).

An equivalent and perhaps more intuitive formulation of regular vari¬

ation on D is given in the next result. Its proof is identical to that of

Theorem 1.15 with obvious notational changes.

Theorem 3.3 Let X = (Xt)t(=[o,i] be a stochastic process with sample

paths in D. Then the following statements are equivalent.

(i) X is regularly varying in the sense of Definition 3.1.

(ii) There exist an a > 0 and a probability measure a on B(Di) such

that, for every x > 0, as u —y oo,

P(|X|00>^,X/|X|00G-)^ _a

p(|xu>u)>* *(•) onB(Dl). (3.5)

Remark 3.4 If (i) holds, then by Theorem 3.2 there exists an a > 0

such that m(uB) = u~am(B) for every u > 0 and B G B(D) and (ii)holds with the same a. If (ii) holds, then (i) holds and m satisfies the

scaling property above with the same a (see the proof of Theorem 1.15

for details).

Definition 3.5 For a stochastic process X satisfying (3.5) we refer to a

and a as the tail index ofX and the spectral measure ofX, respectively.

The two formulations of regular variation on D given above are much

inspired by the formulations of regular variation for random vectors.

Many of them are documented in e.g. Basrak [2].

Remark 3.6 For S G B(D{), let VijS = {x G D : |x|oo > l,x/|x|oo G

S}. It follows from the proof of Theorem 3.3 that the probability mea¬

sure a and the boundedly finite measure m are linked through

Let h : D —>• D or h : D —y Mk be a measurable, positively homogeneous

mapping, i.e. h(Xx) = A/i(x) for A > 0 and x G D. If X is a regularly

varying stochastic process with sample paths in D we may be interested

in the tail behavior of h(X). This is achieved using an analogue of the

Continuous Mapping Theorem for weak convergence. Let D^ = {x G

D : h is discontinuous at x}. Note that D^ G B(D) (see Billingsley [5]

p. 225) and hence also Dh n D G B(D).

Theorem 3.7 (Continuous Mapping Theorem) LetX = (Xt)t<=[o,i]be a stochastic process with sample paths in D. Suppose that there exist

a sequence (an), 0 < an t oo, and a nonzero boundedly finite measure

m on B(D) with m(D\D) = 0 such that, as n —y oo,

nP(a"1X G • ) Â m(-) on B(D).

Let h : D —) D be a positively homogeneous measurable mapping such

that /i_1(.B) is bounded in D for every bounded B G B(D) D D and

suppose m(Dh f\D) =0. Then,

nF(a~1h(X) G •) Âmoh~l(- HD) onB(D).

Moreover, the result holds for mappings h : D —y Mk with the obvious

notational changes.

The formulation of regular variation on D in combination with Theo¬

rem 3.7 allow us to derive the tail behavior of a large class of continuous

mappings of stochastic processes. This will be illustrated in the follow¬

ing sections.

The next theorem gives necessary and sufficient conditions for a stochas¬

tic process with sample paths in D to be regularly varying. Before stat¬

ing these conditions we introduce some notation. For x G D, To C [0,1]and 6 G [0,1] let

w(x,T0) = sup{|xs -xt| : s,t G T0},

w"(x,ô) = sup min{|xt-xtl|,|xt2-xt|}.t1<t<t2,t2-ti<8

Theorem 3.8 Let X = (Xt)t(=[o,i] be a stochastic process with sample

paths in D. Then the following statements are equivalent.

(i) There exist a set T C [0,1] containing 0 and 1 and all but at most

countably many points of [0,1], a sequence (an), 0 < an t °°> and

a collection {mtl...tk ' k G N, ^ E T} of Radon measures with

m(M \Mdk) = 0, and mt nonzero for some t £T, such that

_. dh

nF(a-1(Xt1,...,Xtk)e-)Amt1...tk(-) on B(M \{0}) (3.6)

holds whenever t\,..., tk G T. Moreover, for any e > 0 and r\ > 0,

there exist a ö G (0,1) and an integer no such that

nF(w"(X,Ô) > ane) < m n>n0, (3.7)

nF(w(X,[0,ô))>ane)<r], n>n0, (3.8)

and

nF(w(X, [l-ô, 1)) > ane) <n, n > n0. (3.9)

(ii) There exist a sequence (an), 0 < an t oo; and a nonzero boundedly

finite measure m on B(D) with m(D\D) = 0, such that

nF(a~1Xe -)Âm(-) onB(D). (3.10)

The sequences (an) in (i) and (ii) can be taken to be equal. Moreover, the

measure m in (ii) is uniquely determined by {mtl...tk ' k G N, ti G T}.


Most stochastic processes with sample paths in D and regularly varying

finite dimensional distributions that appear in applications are regularly

varying on D. However, in order to fully understand the conditions of

Theorem 3.8 (i) we find it relevant to study examples of stochastic

processes for which one of the conditions (3.7), (3.8) and (3.9) does

not hold. In Example 3.17 below we construct an additive process

which satisfies all conditions of Theorem 3.8 (i) except (3.9). Roughly

speaking, this additive process is constructed so that for arbitrary small

5 > 0 the probability of extreme jumps within the time interval [1 — 6,1)is too high. Consider also the following example which illustrates a

violation of condition (3.7).

Example 3.9 Let a > 0 and consider independent random variables

Z and V where Z ~ Pareto(o;), i.e. F(Z > x) = x~a for x > 1, and V

is uniformly distributed on [0,1]. Let (Yt)t>o be given by

0 ifte[0,V),

Y=,Z ifte\V,V + l/(2Z)),

1 S0 iî te [V + 1/(2Z),V + 1/Z),z if te [v + i/z,œ),

and let X = (Xt)te[o,i] be given by Xt = Yt for t G [0,1]. Then X

satisfies (with an = n1/") all conditions of Theorem 3.8 (i) except (3.7):for any e > 0 and ö G (0,1)

nF(w"(X,ô) > n1/ae) ~nP(Z> n1/ae) = e~a

as n —y oo.

3.1.1 Proofs

Proof of Theorem 3.7. Let Nh = {x e D : h(x.) = 0} and define

7^ : D\Nh ^Dby

Mx) =

h(x) if xeD\Nh,x if x G D\D.

Then L\ C (Dh n~D)\J (D\D), where D\ denotes the set of points of

D\Nh where h is discontinuous. Take arbitrary bounded B e B(D)with m(h~\dB)) = 0. Since dh~l(B) C h~l(dB) Ui\, we have

m(dh~l(B)) < m(h~1(dB)) + m{L\) = 0. Hence

n F(a~1h(X) G B) = n P(a"1/i(X) G B, h(X) / 0)

= n F(a~lh(X) e B, h(X) / 0)

= nF(a~1X G h~\B) n (D\Nh))

= nF(a~lXeh~1(B))-+ m(h~1(B)).

Hence, by Proposition A2.6.II p. 628 in Daley and Vere-Jones [12],

nF(a~1h(X) G •) Amoh~l(.) on B(D).

However, for every B e B(D), m(h (B)) = m(h~1(B n D)). Hence

nF(a~1h(X) G •) A m o h'1 (-f\ D) on B(D).

The proof for mappings h : D —y Mk is similar. D

Proof of Theorem 3.8. (i) => (ii) Let mn(-) = nF(a~xX e • ). First

we will show that the set {mn} is relatively compact in the w)-topology.

To prove this we will apply Proposition A2.6.IV p. 630 in Daley and

Vere-Jones [12], which says that it is sufficient that the restrictions

{rnn,j} to a sequence of closed spheres S7 f D are relatively compact

in the weak topology. For 7 > 0, let 57 = {x G D : |x|oo > 7}, and for

n > 1, let mn)7(-) = nP(a~1X G • fl S7). We will show that, for every

7 > 0, the family {mn^} is uniformly bounded and that it is relatively

compact in the weak topology.

Take 7 > 0 and ti,...,tk G T with 0 = t\ < • • • < tk = 1 and

ti — U-i < 6, where 6 > 0 is such that nF(w"(X,ö) > anj/2) < r\

for n > no- Then

mnn(D) = nP(|X|oo > a„7)

< n F( max |XtJ > 0^7/2 or w"(X, Ö) > anj/2)l<i<k

< nF( max |Xt,| > anj/2) + nF(w"(X,6) > anl/2)l<i<k

= fn(l)+9n(l)-

By (3.6), (fn(l)) converges to some finite limit as n —y oo and hence

the sequence (fn(l)) is bounded. Moreover, gn(j) < n for n > no, and

clearly gn(j) < no for n < no- Hence, supn>1 mn^(D) < oo, i.e. {mn;7}is uniformly bounded.

Since mn(-) = nF(a~lX G • ) < no for n < no and since a probability

measure P(a~1X G • ) on B(D) is tight it follows by Theorem 15.3 p. 125

in Billingsley [5] that (3.7), (3.8) and (3.9) hold for the finitely manyn preceding no by taking Ö small enough. Hence, we may assume that

no = 1. Note that [7, 00] x K\ G B(D) is compact in D if and only if

K\ is compact in D\. For any n > 0, by (3.7), (3.8) and (3.9), we can

choose Ok such that, if

AK1 = {x G £>i : w"(x, ok) < 1/k},

Ak,2 = {x G £>i : w(x, [0,4)) < 1/fc},

^fc,3 = {x G Di : w(x, [1 - «S*., 1)) < 1/k},

then mn)7([7,00] x (D{\Akj)) < (l/3)n/2fc for every j and n. Let

5 = n^=1 n|=1 Afcj. If K1 is the closure of B, then by Theorem 14.4

p. 119 in Billingsley [5], K\ is compact in D\. Moreover, for every n,

mn>7fD\([7,oo] x K{)) < mre>7([7,oo] x (D{\B))00 3

^ Yl Yl m--7([7,00] x (^A^fc,j))k=lj=l

00

< n^2-fc=n.fc=i

Hence, we have shown that {mTOj7} is uniformly bounded and tight. It

follows from Prohorov's Theorem (Theorem A2.4.I p. 619 in Daley and

Vere-Jones [12]) that {mnjl} is relatively compact in the weak topol¬

ogy. Thus, by Proposition A2.6.IV p. 630 in Daley and Vere-Jones [12],

{nP(a~1X G • )} is relatively compact in the «)-topology. We will

now show that any subsequential w)-limit m satisfies m(D\D) = 0.

By (3.7) and the above argument we can choose u\ and ô such that

nF(w"(X, ô) > anu\/2) < n/2 for every n > 1 (i.e. we may take no = 1

in (3.7)). By (3.6) and Theorem 3.7 (for mappings h : Mk —)> [0,oo))there exist a Radon measure v on i3((0, 00]) with ^({00}) = 0 such that

!/„() := nF(a~1 max |Xtfc | G • ) ^ i/(.) on B((0, 00]).\<i<k

It follows that v has the scaling property described in Theorem 3.2

(the same proof applies with the obvious notational changes). Hence

there exists an a > 0 such that v([x, oo]) = x~av([l, oo]) for every

x > 0. Choose x such that u([x/2,oo]) < n/4. Then there exists n'

such that vn([x/2, oo]) < n/2 for n > n'. Clearly there exists x' such

that vn([x'/2, oo]) < n/2 for n < n'. Hence, with w2 = max(a;,/),

^([^2/2,00]) < n/2 for every n > 1. Hence, with u = max(wi,w2), for

every n > 1,

nPflXloo > anu) < nP(max |XtJ > anu/2)l<i<k

+nF(w"(X,ö) >anu/2)

< n/2 + n/2 = n.

Suppose mn< -^ m. We have just shown that for any r? > 0 there exists

u > 0 such that mn'({x G D : jx^ > u}) < r\ for n' > 1. In particular,

this implies that m„/({x G D : |x|oo > u}) —y 0 uniformly in n' as

u —»• 00. Since Gu = {x G D : (x^ > w} is open and bounded we have

m(Gu) < liminfn'^-oo nin'(Gu) and because of uniform convergence

m(D\D) = lim m(Gu)

< lim liminfmn'(Gu) = liminf lim mn'(Gu) = 0.u—)-oo ro'—>-oo n'—>-oo «—>-oo

Let m and m be two subsequential u)-limits. We will show that m =

m and that 771 is uniquely determined by {mt1...tk ' k G N, £j G T}.Let Tm and T^ consist of those t G [0,1] for which the projection itt

is continuous except at points forming a set of m-measure 0 and fh-

measure 0, respectively. Then, by Theorem 3.7, for rj1?.. . ,rjfc e Tm n

TjnHT,

mo7Tt-?..tfc( • fit*) = mo7rt-U( • nEdfc) = mtl...tk(-) on £(Rd\{0}).

Since Tm, T^ and T each contain all but countably many points of [0,1],the same is true for Tm D T^ D T, in particular Tm D T^ D T is dense in

[0,1]. Moreover, 0,1 G Tm PlT^ PlT. With some minor modifications of

Theorem 14.5 p. 121 in Billingsley [5] one can show that

{7rriltk(H):keN,HeB(Mdk\{0})nMdk,t1,...,tkeTrnnTfhnT}

generates B(D) n D. Hence m and m coincides on B(D) n D and since

m(D\D) = m(D\D) = 0 we have m = m.

(ii) =>• (i) Let Tm consist of those t in [0,1] for which the projection 7rt

from D to Md is continuous except at points forming a set of ra-measure

0. The projections 7ro and 7Ti are continuous and hence 0,1 G Tm. For

t e (0,1), 7Tt is continuous if and only if m({x : xf 7^ *-t-}) = 0. By the

same arguments as in Billingsley [5] p. 124 there are at most countably

many t G (0,1] such that m({x : xf 7^ xt-}) > 0. Then, since m is

nonzero and Tm is dense in [0,1], there exists t G Tm such that mt

is nonzero. Moreover, 7Ttl...tk is continuous except at points forming

a set of m-measure 0 if t\,.. .,tk G Tm. Hence, by Theorem 3.7, for

^li • • • i^k G 1m,

nP(a-1(Xtl,...,Xtfc)G.) = nF(a~1X G ^\Ah{ n Rdfc))

-^ °^U('nRd*) on^(Edfc\{0}).

For t1;..., tk G Tm, let m^...^ (•) = m o tt,"1.^ ( • n Mdk).For n > 1, let mn(-) = nP(a~1X G ). By the scaling property of m,

the set 5„ = {x 6 D : |x|oo > u] is an m-continuity set for every u > 0.

Hence, mn(Su) —y m(Su) = w_am(5'i) for every w > 0. Choose w such

that u~am(Si) < n/4. Then there exists n\ such that mn(Su) < n/2for n > n\. By Proposition A2.6.IV p. 630 in Daley and Vere-Jones

[12], for every 0 < 7 < u < 00, {mn( • D {x G -D : |x|oo G [7,^]})} is

relatively compact in the weak topology on D. Since {x G D : |x|oo G

[7, m]} C .D\{0} and on this subspace the subspace topologies (of D

and D) coincide it follows that {mn( n {x G D : |x|oo G [7, w]})} is

relatively compact in the weak topology on D. Hence, by Theorem 15.3

p. 125 in Billingsley [5], for any e > 0 and n > 0 there exist ö G (0,1)and integer n ane, (X^ G an[j,u]) < n/2, n > n2,

nF(w(X, [0,(5)) > ane, JX^ G an[j,u]) < n/2, n > n2,

and

nP(w(X, [1- 6,1)) > ane, (X^ G an[7,w]) < n/2, n > n2.

In particular the three inequalities above hold, with n/2 replaced by n

and n2 replaced by no = max(ni,n2), for u = 00 and 7 < e/2 and for

such 7 they coincide with (3.7), (3.8) and (3.9). D

3.2 Markov processes with asymptotically

independent increments

In this section we will study Markov processes with increments that

are not too strongly dependent in the sense that an extreme jump does

not trigger further jumps or oscillations of the same magnitude with a

nonnegligible probability. We will derive surprisingly concrete results

for such Markov processes (see Theorem 3.12 and 3.18) which will prove

very useful when used in combination with Theorem 3.7 (see e.g. Ex¬

ample 3.20 and 3.21).

Let (Xf)iG[0)i] be a Markov process on Md with transition function

Ps,t(x, B). For r > 0 and 0 < u < T < 1 define

arjT(u) = sup{PSjt(x, B^r) :xG Md and s,t e [0,T],t- s G [0,u]}.

Note that if the random vectors Y and Y are independent and, for

some sequence (an), 0 < an f oo, and Radon measures m and m with

ra(Ed\Rd) = m(Md\Ed) = 0 we have

nF(a~1Y e •) A m(-) and nP^YG-)^^) on £(Rd\{0}),

then, by Theorem 1.28,

nP(a"1(Y + Y) G -)^m(-)+m(-) onB(Md\{0}), (3.11)

i.e. the limit measure of the sum is the sum of the limit measures.

Independence of Y and Y is not necessary for (3.11) to hold. For

Markov processes the much weaker condition o;T.)i(l) —y 0 as r —y oo is

sufficient (with Y and Y representing two nonoverlapping increments)as shown in the following lemma.

Lemma 3.10 Let (Xt)t<=[o,i] be a Markov process on Md such that

Oir,i(l) —> 0 as r —y oo. Fix arbitrary s,t G [0,1] with s < t. Let (an)be a sequence with 0 < an t oo, and let ms, mt and p be Radon mea¬

sures on B(Md\{0}) with ms(Md\Md) = mt(Md\Md) = p(M^\Md) = 0.

3.2. Asymptotically independent increments 79

Consider the following statements.

nF(a~1Xs G • ) A ms(-) on B(m\{0}), (3.12)

nP(a"1Xf G • ) ^ mt(-) on B(Md\{0}), (3.13)

nF(a-1(Xt-Xs)e-) ^ p(-) on B(M.\{0}). (3.14)

// any two of the above three statements hold, then the third also holds

and the limit measures are related through mt = ms + p.

Lemma 3.10 justifies the following choice of terminology.

Definition 3.11 A Markov process (Xt)t^[o,i] on Md is said to have

asymptotically independent increments if av,i(l) —> 0 as r —y oo.

It turns out that for a strong Markov process (see Remark 2.6) with

sample paths in D and asymptotically independent increments we can

obtain sufficient conditions for regular variation on D, which are easier

to verify than the general conditions of Theorem 3.8.

Theorem 3.12 Let X = (Xf)fG[o7i] be a strong Markov process with

sample paths in D such that ar,i(l) —y 0 as r —y oo. Suppose there

exist a set T C [0,1] containing 0 and 1 and all but at most countably

many points of [0,1], a sequence (an), 0 < an t oo, and a collection

{mt : t e T} of Radon measures on B(M.d\{0}), with mt(Md\Md) = 0

and with mi nonzero, such that

nF(a~1Xt e-)^ymt(-) on B(Md\{0}) for every t G T, (3.15)

and such that, for any e > 0 and n > 0 there exists a 6 > 0, ô G T,

1 — ô G T such that

ms(Bc0je)-m0(Bc0je)<r] and m1(Bc^e) - ml.s(Bc0^) < n. (3.16)

Then there exists a nonzero boundedly finite measure m on B(D) with

m(D\D) = 0, such that, as n —y oo,

nP(a"1X G • ) Â m(-) on B(D).

Moreover, the measure m is uniquely determined by {mt : t G T}.

Remark 3.13 By Theorem 11.1 p. 59 in Sato [38], a sufficient condition

for a Markov process (Xt)t>o on Md to have a version in D is that

limaeT(u) = 0 for any e > 0 and T > 0.u4-0

Theorem 11.1 considers Markov processes with fixed starting points,

however this requirement can be dropped as seen from the proof. Note

that the above condition implies stochastic continuity.

Remark 3.14 We have chosen to formulate the above theorem and

results below for strong Markov processes, since it is convenient to use

the strong Markov property in some of the proofs. We could however,

instead of the strong Markov property, have assumed that we have just

the Markov property and that, for every e > 0, na2e x(l) —y 0 as

n —y oo.

The following result, for additive processes, was given as Theorem 2.11

in Chapter 2. However, the result and the proof apply in our more

general setting.

Theorem 3.15 Let X = (Xt)t<=[o,i] be a strong Markov process with

sample paths in D such that ar;i(l) —y 0 as r —y oo. Then the following

statements are equivalent.

(i) There exist T C [0,1] such that 0,t eT and such that [0,1]\T is

at most countable, a sequence (an), 0 < an t °°? and a collection

of Radon measures {ps : s G T} on B(M \{0}) such that for every

s eT, ps(M \Md) = 0, pt is nonzero, and

nF(a~1Xs e ) A ps(-) on B(m\{0}). (3.17)

(ii) There exist a sequence (an), 0 < an t °°> and a nonzero Radon

measure p on B([0,1] x (Ë*\{0})) with p([0,1] x M^XM*1) = 0 such

that

nP({(s,Xs) :0<s< 1} D (an * • ) / 0) A p(-) (3.18)

onß([0,l]x (Md\{0})).

Furthermore, the sequences (an) in (i) and (ii) can be taken to be equal

and then the measures ps and /l([0,s] x • ) coincide on B(M \{0}) for

every s G [0,1].

Remark 3.16 As seen from the proof of Theorem 2.11, if nF(an1Xs G

• ) A- ps(-) on £(Ëd\{0}) for every seT, then nP(a"1Xs G • ) A ps(-)

on #(Ëd\{0}) for every s G [0,1].

The following example shows that one can find an additive process

X = (Xt)t(=[o,i] which has a regularly varying graph (i.e. satisfies the

conditions of Theorem 3.15 (ii)) but which is not regularly varying on

D (i.e. satisfies neither the conditions of Theorem 3.12 nor those of

Theorem 3.8).

Example 3.17 Let v be a probability measure on [0,1] x [1, oo) given

byoo

v(dt x dx) = ^2kl{1_1/k^](t)dtl[kjk+i)(x)ax~a~ldx.k=i

Let £ be a Poisson random measure with intensity measure v and let

X = (Xt)t£[o,i] be a stochastic process given by

Xt = / xt;(ds x dx).J[0,t]x[l,oo)

Then, by the same argument as in Example 2.14, X is an additive

process and, for every t, vt(-) = ^([0,£] x •) is the Levy measure of Xt.

Take u > 0 and note that

/»oo

n£([0,1] x (n1/au, oo)) = n / ax'^dx = u~a,Jn1/au

and for t < 1 and n large enough,

nv([0,t] x (n1/au, oo)) = 0.

Hence, for every t G [0,1],

nut(n^a -)^pt(-) on£((0,oo]),

where

r o ift<i,

I SBocx-a~ldx ift=l.

Fix e > 0. For every 5 G (0,1) and large enough n,

nF(w(X,[l-5,l)) >n1/ae)

= nP( sup \XS-Xt\ >n1/ae)s,te[i-6,i)

>nP(C((l-(n1/ae)"1,l) x [n1/ae,oo)) > 0)

= n(l - exp{-£((l - (n1/ae)~\ 1) x [n1/ae, oo))})

> \nv((l - (nllae)-\ 1) x [nê, oo))Li

> \nv((l ~ ([n1/ae] + 1)"\1) x [[nê] + 1, oo))

1oo

= 0" E (k~a ~ (k + l)~a)fe = [n1/ae] + l

= ln([nâe] + 1)"° - ±e~a

as n —y oo. Hence, for n < ê_a there exists no ô G (0,1) such that

nF(w(X, [1 — J,l)) > n1/ae) < n for all large enough n. Hence, by

Theorem 3.8, X is not regularly varying on D!

It turns out that a regularly varying strong Markov process with sample

paths in D and asymptotically independent increments has a very simple

extremal behavior. In this case the process reaches a set far away from

the origin by making at most one jump to that set (it might start there

at time 0 since we allow for a regularly varying starting point) and the

process essentially stays constant before and after the jump. This is

formalized in the next theorem. Let

V = {x G D : x = yl[Vil],v G [0, l],y G Ed\{0}}.

Theorem 3.18 Let X = (Xt)t(=[o,i] be a strong Markov process with

sample paths in D such that arj\(l) —y 0 as r —y oo. Suppose there exist

a sequence (an), 0 < an t oo, and a nonzero boundedly finite measure

m on B(D) with m(D\D) = 0 such that, as n —y oo,

nP(a"1X G • ) Â m(-) on B(D).

Then m(Vc) = 0. Moreover, there exist an a > 0 and a probability

measure a on B(D\) such that, for every x > 0, as u —y oo,

P(|XU>n,,X/|XUG_O^_M0on

P(|X|00>w)

«;i£/i cr({x G Di : x = yl^i], v G [0, l],y G §d_1}) = 1. Moreover, on

a({y:eDl : x = yl[ü;1], v G [0,1], y G • })

coincides with the spectral measure ofXi.

For Levy processes we can be even more explicit.

Example 3.19 Let X = (Xt)t<=[o,i] be a Levy process on Md. Suppose

there exist a sequence (an), 0 < an t oo? and a nonzero Radon measure

mi with mi(ld\Md) = 0 such that

nP(a"1Xi G • ) ^ mi(-) on ß(Rd\{0}).

Since X has stationary and independent increments, nP(a~1Xi G ) A

imi(-) on jB(M \{0}) for every t G [0,1]. Hence, combining Theorems

3.3 and 3.12 gives, for every x > 0, as u —y oo,

P(W,>^,x/|x|,6.)^P(|X|oo > U)

where a > 0 is the tail index of Xi and it follows that

a(-) = F({zi[VA](t),te[o,i]}e-),

where Z and V are independent, the distribution of Z is the spectral

measure of Xi and V is uniformly distributed on [0,1]. The random

vector Z is the direction of the big jump and V is the time of the big

jump. See also Figure 3.1.


The following two examples illustrate the usefulness of Theorem 3.18

in combination with Theorem 3.7. Compare with the rather technical

proofs needed to prove the corresponding results in the special case of

additive processes (Theorems 2.11 and 2.16).

Example 3.20 Let X be a strong Markov process with Xq = 0 sat¬

isfying the conditions in Theorem 3.18 and let h : D —y Md be defined

by

/i(x) = ( sup x\ ,..., sup x\ '\.MG[0,1] *G[0,1] '

Then it is straightforward to show that h satisfies the conditions of

Theorem 3.7. Hence,

nF(a~1h(X) e ) A m o h~\- n Md) on B(m\{0})

and for B e B(m\{0}), with Mf = [0, oo)d,

mo/r1(ßnRd) = m({xEÖ:/i(x)eßnId}nV)= m({x G D : x = yl[Vil]iv G [0,1], y G B n Md+})= m^-^B nR|)nv)= m^-^BDMJl))= mi(ßnKj),

where mi is vague limit of (nP(a~1Xi G )).

Example 3.21 Let X be a strong Markov process with X0 = 0 sat¬

isfying the conditions in Theorem 3.18 and let h : D —y Md be defined

by

Mx) = (jf x^dt,..., / x[d)dt

Then it is straightforward to show that h satisfies the conditions of

Theorem 3.7. Hence,

nF(a~1h(X) e ) -^ m o h~\- n Md) on B(m\{0})

and for B e B(m\{0})

mor1(ßni(l)= m({x G D : h(x) G^nl^nV)= m({x G D : x = yl[Vjl]lv G [0,1], y(l - v) G B n Md})

= m({x G £> : xf G B n Md some £ G [0,1]} D V)J. L

= m({x e D :xt e B n Md some £ G [0,1]}).J. L

In particular, if (Xt)t£[o,i] is a Levy process, then the last expression

reduces to (see also Theorem 2.16)

/ mi(—^—BnMd)ds = mi(BnMd) f (1 - s)ads = —mi(5),Jo 1~s Jo a + !

where mi is vague limit of (nP(a~1Xi G ))

3.2.1 Proofs

Proof of Lemma 3.10. Fix a relatively compact B e B(M \{0}).Then there exist r > 0 such that B C Bqt. Note that the scaling

property implies that sets of the form Bq r,r > 0, are always ms-, mt-

and //-continuity sets.

Suppose that (3.12) and (3.13) hold. We first show that {nP(a"1(Xi -

Xs) G • )} is vaguely relatively compact. We have

supnF(a~l(Xt - Xs) e B) < supnF(a~1(Xt - Xs) G Bc0 r)

< supnP(a"1Xs G B^ r/2)+supnF(a~1Xt G ßgr/2) < oo,n>l

'

n>l'

since {nP(a~1Xs G • )} and {nF(a~lXt G • )} are vaguely relatively

compact. Hence {nP(a~1(Xf — Xs) G • )} is vaguely relatively compact.

By essentially the same argument it follows that if (3.12) and (3.14)

hold, then {nP(a~1Xt G • )} is vaguely relatively compact, and if (3.13)and (3.14) hold, then {nP(a~1Xs G • )} is vaguely relatively compact.

Suppose that (3.12) and (3.13) hold. Let p be a subsequential vague

limit such that

n'F(a-}(Xt-Xs)e-)^p(-).

Fix ei > 0, Ê2 > 0 and a relatively compact B G B(M \{0}) with

ms(dB) = p(dB) = 0. We have

n'F(a^(Xs,Xt-Xs)eBc0jeixBc0j£2)= n'F(a^Xs e Bc0jei)F(a-ï(Xt - Xs) G B^ \ a~^Xs G B^J

v

v'v

v'

-0.

Since ms(M \Md) = p(M \Md) = 0 we may without loss of generality

assume that B n Md / 0. Then,

n'F(u;(XS)Xt-Xs)eßx%2)= n'Pfc1^ G B)(l - F(a~}(Xt - X.) G Sg>£a | a~}Xs G S))

vv

' >

v'

^SCB) <aa ,e2,i(l)->-0

^ras(£).

Clearly,

n'P(a-1(Xs,Xf-Xs) G B0,eixB) < n'F(a-^(Xt-Xs) e B) ^ p(B).

Set 7 = infxGBnEd |x|. Then

n/P(a-/1(Xs,Xt-Xs)G50,ei x B)

= n'F(a-^(Xt-Xs)eB)- n'F(a^Xs e Bc0^)F(a^(Xt -Xs)eB \ a~}Xs G B^J

>n'F(a-^(Xt-Xs)eB)- n'F(a-^Xs e B^ei)F(a-?(Xt - Xs) G B%„ \ a~^Xs G B^J

vv

/Nv

'

->-m*(Bo,ei) <aan/7,i(l)^0

->/*(£)

It follows that n/P(a-/1(XB,Xt-Xa) G • ) A p(-) on #(R2c\{0}), where

/2 is a Radon measure which concentrates on ({0} x Md) U (Md x {0}).

Hence

ri F(a~ï(Xs + Xt-Xs)e-)^ /j((x, x) : x + x G • ),

where

//((x,x) :x + xG •) = /2((x,0) :x + 0 G • )+/2((0,x) : 0 + ÏG • )

= ms(-)+p(-).

However, n'F(a~^(Xs-\-Xt — Xs) G • ) -4- mt(-) and hence p = mt—ms.

Since this is true for any subsequential vague limit of (nP(a~1(Xi —

Xs) G • )) it follows that

nP(a"1(Xt - Xfl) G • ) A mt(-) - ms(-).

Suppose now that (3.12) and (3.14) hold. By the same arguments as

above, replacing n' by n, it follows that

nP(a"1XfG-) = nF(a-1(Xs + Xt-Xs)e-)

A /2((x,x) : x + x G • ) = ms(-) + p(-).

Suppose now that (3.13) and (3.14) hold, and let ms be a subsequential

limit such that n' F(a~?Xs G • ) -^ ms(-). By the same arguments as

above it follows that

ri F(a~}(X3 + Xt-Xs)e-)^ /2((x, x) : x + x G • ) = m3(-) + //(•)

However, n' F(a~}(Xs-\-Xt — Xs) G • ) A rnt(-) along every subsequence

(n') so we must have ms = mt —

p.

To prove Theorems 3.12 and 3.18 we need a couple of technical lemmas.

For e > 0, positive integer p and M C [0,1] we say that an element

x G D has e-oscillation p times in M if there exist to, , tp G M with

to < • • < tp such that |x^ — xij_11 > e for i = 1,... ,p. Let

B(p, e, M) = {x G D : x has e-oscillation p times in M}.

The following lemma is an immediate consequence of Lemma 2 p. 420

in Gihman and Skorohod [20].

Lemma 3.22 Let X = (Xt)iG[0,i] ^e a Markov process with sample

paths in D. If for e > 0 and 0 < Ti < T2 < 1 the quantity ctej^Ti2^T2 —

Ti) is less than 1, then

®e/4,T2(T2-Ti)F(XeB(l,e,[Ti,T2]))<

1 - Û!e/4,T2 (Î2 -Ti)

Proof. First,

P(XG5(l,e,[T1,T2])) = P( sup |X,-Xs|>e)s,te[TuT2]

< F( sup |Xs-XTl| >e/2),aG[Ti,T2]

and, by Lemma 2 p. 420 in Gihman and Skorohod [20],

P(|Xt -Xt I > e/4)P( sup |x. - XTJ > e/2) <-^_Zk Wt\-

Finally, P(|XTa - XTl| > e/4) < o;e/4)T2(T2 - 7\) from which the con¬

clusion follows. D

Lemma 3.23 LetX = (Xt)fG[0;i] be a strong Markov process with sam¬

ple paths in D such that ct;r)i(l) —y 0 as r —y oo. Suppose there exist

a sequence (an), 0 < an t °°; and Radon measures mo and mi on

#(ld\{0} with mo(Md\Md) = mi(Md\Md) = 0 such that

nF(a~1X0 e • ) 4m0(') and nF(a~1Xi e • ) ^y mi(-)

on B(Md\{0}). Then, for every e > 0, nP(X G anB(2,e, [0,1])) -y 0 as

n —y oo.

Proof. Fix an arbitrary e > 0 and let rn = inf{t : \Xt — X0| > ane/2}with the convention inf 0 = oo. Then

nP(Xean5(2,e,[0,l]))

< nE(l{rn<1}E^X- (lB(l,a„e,[r„,l])(X)))< nE(l{rn<1}aaTie/4)1(l)/(l - aane/4>1(l)))

0!ane/4,l(l)nP( sup |Xt -X0| > ane/2)

by combining Lemma 3.22 and the strong Markov property. Moreover,

by combining Lemma 2 p. 420 in Gihman and Skorohod [20] and Lemma

3.10,

W iv y I >> /o\ <rnF(\Xi-Xo\>ane/4)

nP( sup |Xt-

X0|> ane/2) < t—

*G[0,1]-L -«ane/4,lUJ

" ml(BO,e/d-m0{BCQ,e/4),

as n —y oo, from which the conclusion follows. D

Proof of Theorem 3.12. Fix s,t eT with s < t. We will show that

there exists a unique vague limit msj such that nF(a~1(Xs, Xt) G • ) A

mSjt(-)- By repeating the procedure one can then show that, for any k G

N, there exists a unique vague limit mtl,...,tk, with mtlj...jtk(M \Mdk) =

0, such that nP(a"1(Xfl,..., Xifc) G j A mtu...,tk (•) ifh,...,tke T.

By Lemma 3.10,

nP(a"1(Xt - X8) G • ) A mt(-) - ms(-).

Clearly, there exist unique vague limits (Radon measures) mSjS and

m on £(R2d\{0}) with mS:S(M2d\M2d) = m(M2d\M2d) = 0 such that

nP(a"1(Xs,Xs) g • ) A m^(-) and nP(a"1(0,Xi - Xs) G • ) ^ m(-)

on B(M \{0}). By arguments similar to those in the proof of Lemma

3.10,

nP(a"1(Xs,Xi)G-) = nP(a"1((Xs, X5) + (0,Xt - X5)) G • )

A m8>8(-)+m(.)=:ma>t(-) on ,B(R2d\{0}).

Note that, by Lemma 3.23,

nF(w"(X,ô) > ane) <nF(Xe anB(2, e, [0,1])) - 0

as n —y oo. Hence, for any positive e and n there exists an no such that

nF(w"(X,ô) > ane) < n for any ô G (0,1) if n > no- Hence condition

(3.7) of Theorem 3.8 holds.

It remains to show that conditions (3.8) and (3.9) also hold. Fix arbi¬

trary e > 0 and n > 0. By Lemma 2 p. 420 in Gihman and Skorohod [20],

nF(w(X,[l-Ô,l))>ane) < nP( sup |X* - Xi_Ä| > ane/2)te[i-s,i]

<wP(|Xi-Xi_f| >ane/4)

l-Û!ane/4,l(<5)

By Lemma 3.10, for 1 - 5 G T

Jim^nPdXi - Xi_*| > ane/4) = mi(Bê/A) - mi-ô(BcQe/A).

Hence, by (3.16) there exists a <5 > 0, 1 — ô eT such that

r »/ fY f) niw wrwP(|Xi - Xi_j| > qwe/4)

lim sup nF(w(X, [1—

o, 1))>

ane)< hmsup

ri->-oo n-ôo 1 — aane/4,l (")

= ml(B0,e/4) ~ ml-s(Bo,e/4) < V,

and it follows that (3.9) holds. That (3.8) holds is shown by an almost

identical argument. The conclusion now follows by Theorem 3.8. D

Proof of Theorem 3.18. First note that B(2, e, [0,1]) is open

and, by Lemma 3.23, rimmf^«, nP(X G anB(2, e, [0,1])) = 0. By

assumption, liminfn_^00nP(X G anG) > m(G) for every open bounded

G e B(D). Hence m(B(2, e, [0,1])) = 0. Since e > 0 was arbitrary

it follows that m(B(2, e, [0,1])) = 0 for every e > 0 and hence also

m(Ue>0,ee®B(2, e, [0,1])) = 0. Since

Ue>0,eGQ5(2,e,[0,l])=(ö\JD)UVc,

it follows that m(Vc) < m(Ue>0,eGQ-B(2,e, [0,1])) = 0. Moreover, by

Theorem 3.3, there exist a > 0 and a probability measure a such that

(3.5) holds. Furthermore, by Theorem 3.7,

nP(a"1X1 G •) AmoTTÔ on 5(Rd\{0}),

which holds if and only if there exists a probability measure o~i on

B(Sd~x) such that, for every x > 0, as u —y oo,

p(|Xi| > u)>* *,(•) onB(S )

holds, and &i is given by

,.mo7rf1({xGld\{0} : |x| > l,x/|x| G • })

al-\') = ZZÂ

moTrf^jxGE \{0} : |x| > 1})

3.3. Filtered Levy processes 91

We have

P(|X|00>gn,X/|X|00G-)

F(\X\00>an)

_

nP(g~1X e{xeD : |x|qq > 1,x/|x|qq G • })

nP(anXX e{xeD: |x|oo > 1})

_^m({x G £> : Ixloo > 1,x/|x|qq G • })

m({x G o : [XU > 1})

which necessarily is equal to a(-). Moreover,

<r({x Gfli:x = yl[Vil]iv G [0,1], y G S^1})

_

m({xe D : [xloo > 1} n V)

m({x G D : |x|oo > 1})

m({x G Ö : Ixloo > 1})1

ra({xG£>: Ixl«, > 1})

and

m(7T1~1({x G Ëd\{0} : |x| > l,x/|x| G • }))

m(7rr1({xGËd\{0}:|x|>l}))

m(7r1~1({xGRd\{0}: |x| > l,x/|x| G • }) fl V)

m(7rr1({x G ld\{0} : |x| > 1}) n V)

m({x6D:x = ylM,t)G [0,1], |y| > 1,y/|y| G • })

m({x6D:x = ylM,üe [0,l],|y|> 1})

_

m({xG £> : |x|qq > 1,x/|x|qq = yl[t,,i],^ G [0,1],y G • })~~

m({x G D : |x|oo > l}n V)= <t({xG£>i :x = ylKl],^G [0,l],yG-}).

The conclusion follows. D

3.3 Filtered Levy processes

In this section we will give another application of regular variation on

D by studying asymptotics of stochastic processes Y of the type

Yt= ( f(t,s)dX8, te [0,1], (3.19)Jo


where X is a regularly varying Levy process with sample paths of finite

variation. The idea here is that X is a regularly varying strong Markov

process satisfying arji(l) —y 0 as r —y 0, and that extremes for the

process Y are caused by one big jump in the process X. It turns out

that Y and Hf(X), where Hf : D —y D is defined below, have the same

regular variation limit measure and that Hf is sufficiently regular so

that Theorem 3.7 can be applied (we only need that Hf is positively

homogeneous on V - the set of step functions with one step). In this

way we can show that the process Y is regularly varying. Furthermore,

and equally important, we are able to explicitly compute the spectral

measure of such processes. In doing so we provide a natural way to

understanding the extremal behavior of such filtered regularly varying

additive processes. As a concrete example we will compute the spectral

measure of an Ornstein-Uhlenbeck type process driven by a regularly

varying Levy process (Example 3.25). Note that finite variation of the

sample paths of X allows us to define the integral in (3.19) in a pathwise

sense. As in the previous section, let

V = {xefl:x = yl[w,i], v G [0, l],y G Ed\{0}}.

Moreover, let Vo — V U {0} and let do denote the so-called Ji-metric

(see Billingsley [5] p. 112). For x G D define

M(x) 4 {z G Vo : d0(z,x) = inf{do(z,x) : z G V0}},

i.e. M(x) consists of the step functions in D with one step that are

closest to x. Note that for every xGDwe have M(x) ^ 0, see Lemma

3.26 below for details. Define \Ü : D —y Vq such that for x G D we take

\P(x) to be a unique element of M(x) chosen according to some arbitrary

criteria (e.g. of the elements of M(x) with earliest jump choose ^(x)as the one with biggest jump). For a nonzero and continuous function

/ : [0, l]2 - M define hf : V0 -^ D by

Mx)* = f f(t,s)dxs, te [0,1].Jo

Finally, define Hf = hf o\ü. Note that Hf is in general not continuous.

However, it is continuous on Vo, and this is sufficient when considering

integrators whose regular variation limit measure concentrates on V C

V0.

Theorem 3.24 Let X = (Xt)te[o,i] be a Levy process on Md. Suppose

that there exist a sequence (an), 0 < an f oo; and a nonzero boundedly

finite measure m on B(D) with m(D\D) = 0 such that, as n —y oo;

nF(a~lX G • ) A m(-) on B(D).

For a nonzero and continuous function f : [0, l]2 —y M, define the process

Y = (Yt)te[o,i] by Yt = f0 f(t, s)dXs. Then Y has sample paths in D

and, as n ^ oo,

nF(a~1Y e •) AmoHj1(- DD) onB(D).

To illustrate Theorem 3.24 we will now compute the spectral measure of

an Ornstein-Uhlenbeck type process driven by a regularly varying Levy

process.

Example 3.25 Let X = (Xt)te[o,i] be a Levy process on Md with sam¬

ple paths of finite variation. Necessary and sufficient conditions for

having sample paths of finite variation are that the generating triplet

(A, v, 7) satisfies A = 0 and either (i) j/(Rd\{0}) < oo or (ii) j/(Rd\{0}) =

oo and J|xi<lx^0 WK^x) < oo (see Sato [38] p. 140). Suppose there

exist a sequence (an), 0 < an t oo, and a nonzero Radon measure mi

with mi(ld\Rd) = 0 such that

nP(a-1X1 G • ) A mi(-) on ß(Rd\{0}).

Since X has stationary and independent increments, this implies that

nF(a;1Xt G • ) A tmi(-) on £(Rd\{0}) for every t G [0,1]. Let

Y = (Yt)te[o,i\ be an Ornstein-Uhlenbeck type process driven by X,

given by

Yt = [ e-d{t-s)dX8, 9>0, te [0,1Jo

Hence, by combining Theorems 3.3, 3.12 and 3.24, for every x > 0, as

u —y oo,

^\\ *oo> UX, Ï/ I

t» t ' j w —a / •> z?/n\>X a(-) «.BID,),

where a > 0 is the tail index of Xi and it follows that

a(.)=F({Ze-^-vh[VA](t),te[0,l]}e-),

where Z and V are independent, the distribution of Z is the spectral

measure of Xi and V is uniformly distributed on [0,1].

For the proof of Theorem 3.24 we will need the following results.

Lemma 3.26 M(x) ^ 0 for every x G D.

Proof. Fix x e D. For e G (0,1], define

Ke = {{y,v) :y G B0iSuPte[oA]lxtl,ve [0,1-e]}.

For some e G (0,1] we have

inf{d0(z,x) : z G V0} = inf{rf0(yl[v,i],x) : (y,v) G Ke).

Since Ke is compact and since (y, v) \-^y do(yl[w,i], x) is continuous there

exists (y*, v*) G Ke such that

mf{d0(y±[t,,i],x) : (y, v) e Ke} = d0(y*l[uV], x),

i.e. M(x) is nonempty. D

Lemma 3.27 Hf = hf o ^ is continuous on Vo-

Proof. We first show that \I/ is continuous on Vo and then that hfis continuous. Take xq G Vo and let (xn) be a sequence in D such

that do(xTC,xo) —y 0 as n —>• oo. By construction, <io(^(xn),xn) <

cfo(xo,xn). Since \I/(xo) = xo we have

d0(^(x„),^(x0)) = do(^(xn),x0)

< d0(^(xn),xn)+ c?o(xn,xo).

Hence do(^(xn), ^(xo)) —y 0 as n —y oo which proves the first claim.

We now show that hf is continuous. It is sufficient to show that hf

is continuous on Vo C D equipped with the Skorohod metric since this

metric and the Ji-metric are equivalent (see Billingsley [5] p. 114). Take

x G V and let (xn) be a Vo-valued sequence such that xn —y x. This

implies that there exists no such that xn G V for n > no and hence

we can without loss of generality assume that xn G V for every n.

Then there exist y,yn,v,vn such that xn = ynl[Un,i] and x = yl[v,i].Moreover, there exists a sequence (An) of strictly increasing continuous

mappings of [0,1] onto itself satisfying suptGr0 -n \Xn(t) — t\ —y 0 and

sup |ynl[ün,i](An(*))-yl[t,,i]WI->0 asn^oo.

te[o,i]

First we show that xn —y x implies that yn —y y and vn —y v. Since

sup \ynl[vn,i](K(t)) - yl[«,i] W| > |yn - y|,te [o,i]

it follows that yn —y y. Suppose that vn -ft v. Then there exists e > 0

such that limsup^^^ \vn — v\ > e. Since suptGr01i \Xn(t) — t\ —y 0 and

yn —>• y, this implies that

limSUp SUp |ynl[r,„,l](AnW)-yl[t;,l]WI ^ MAn-^oo te [0,1]

which is a contradiction. Hence vn —» v. We may now proceed to show

that xn —y x implies hf(x.n) —y hf(x). Indeed

sup

*e[o,i]

/»A„(i) pt/ f(\n(t),s)dXn(s)- / f(t,s)d*(s)Jo Jo

< SUp \f(\n(t),Vn)(ynl[Vn,l](\n(t))-yl[v,l](t))\te[o,i]

+ sup \(f(\n(t),vn)-f(t,v))yl[Vil](t)\*G[0,1]

• 0.

*G[0,1] *G[0,1]


Since / is uniformly continuous on [0, l]2, supfGr0jli \Xn(t) — t\ —>• 0 and

vn —y v it follows that

sup \f(Xn(t),vn)- f(t,v)\ ->0,*e[o,i]

i.e. hf is continuous on V. Finally, if xn —y 0, then clearly hf(x.n) —y 0.

Hence hf is continuous on Vo-

Lemma 3.28 If B G B(D) is bounded inD, then Hj1(Br\D) e B(D)is bounded in D.

Proof. We will show that for each r > 0 there exists an r = r(r, /) > 0

such that sup/G[0)1]|/0'/(*5s)d*(x)a| > r implies suptG[0)1] |xt| > r,

i.e. that HJ1(Bqt) C Bq ~,from which the conclusion follows. Fix

r > 0 and suppose that

sup | / /(i,s)d#(x)fl| >r.

*G[0,1] Jo

Then tf (x) = yl[„,i] with y G Rd\{0} and v G [0,1). Hence

te

which implies

sup | / f(t,s)d^(-x)s\ = sup \f(t,v)\\y\>rg[o,i] Jo te[o,i\

|y|>suPu,ve[o,i]\f(u,v)\'

Since, by construction of \I/, supfGr01] |x/| > |y|, we have

i ir

sup xt > —

-.

te[o,i] suPu,ve[o,i]\f{uiv)\

D

Lemma 3.29 Y has sample paths in D.

Proof. By assumption there exists fi'cfi with F(Q') = 1 such that

for each uj G Q', X(uj) G D and has finite variation. For such uj we also

have, since / is continuous on [0, l]2 and hence also uniformly continuous

on [0, l]2,

limv\t

/ (f(t,s)-f(v,s))dX8(u)'[0,v]

<lim sup \f(t,s)- f(v,s)\FV(X(u);[0,t]) = 0,vt* 8e[o,t]

where FV(g;T) denotes the total variation of g on T C [0,1]. Hence,

for uj e Q',

\im(Yt(uj) - Yv(uj)) = lim ( / /(*, s)dXs(uj) - [ f(v, s)dXs(u)VV vft \J[o,t] J[0,v]

= lim f (f(t, s) - f(v, s))dXs(uj) + lim f f(t, s)dXs(uj)vt* J[o,v] vtt J(vj]

= 0 + f(t,t)(Xt(uj)-Xt-(u)),

since X(uj) is right-continuous with left limits. Similarly,

\im(Yv(uj) - Yt(uj)) = lim ( / f(v, s)dXs(uj) - [ f(t, s)dXs(u)H* vit \J[o:V] j[o,f]

= lim f (f(v, s) - f(t, s))dXs(co) + lim f f(v, s)dX8(uj)H* J[0,t] vit J(t,v]

= 0 + f(t,t)(Xt+(uj)-Xt(u)) = 0.

Hence Y (a;) is right-continuous with left limits. D

Proof of Theorem 3.24. By Lemma 3.29, Y has sample paths in

D. Since m vanishes on Ve and, by Lemma 3.27, Hf is continuous on

V, it follows as in Theorem 3.7 that

nP(i7/(a~1X) G )AmoHj1(- DD) on B(D)

(here we do not need positive homogeneity of Hf). We now show that

this implies that nP(a~xY G • ) 4> m o Hjl( DD) on B(D), from

which the conclusion follows. Without loss of generality we assume

that supwvG[01] \f(u,v)\ — 1 (to avoid having to introduce additional

constants). For y G D and r > 0 let Byr = {z G D : d0(y,z) < r}.Fix arbitrary x G -D\{0} and 0 < e < ô < 7 with 7 + ô < cfo(x, 0)

and e < 7 — ô. Moreover, 7 and ô are chosen so that 5X,7, SXj7_j and

BXjl+s are mo H, -continuity sets. Let Xn be given by

X?^/*l«8....(AX.)dX.*J u

(where £0)a„e = {* G Md : |x| < ane}). Then Xn and Xn = X-XTC are

independent Levy processes for all n. Hence we can write Y — Yn+Yn

where Yn and Yn are independent and Y — J0 /(£, s)dX and Y —

/0'/(M)dX£. Note that

nP(a-1YG5X;7) > nF(a~1Yn e B^^s^'1^ e B0jS)

= nP(a-1Yn G B^_s)F(a-1Yn G So,*)

and that

nP(a-1YGSx,7) < nF(a~1Yn G 5x,7+(5, a~lYn G So,*)

+nP(a-1Y-Gß^)- nP(a-xYn G 5x,7+(5) Pfe1^ G B0,s)

-^nF(a-1Yn e Bc0jS)

Hence if we show that

nP(a-1Y"G^,)^0,

nF(a~1Yn G Sx>7_*) - m(Hj\B^_ô) n Sg>e),

nP(a-xY" G 5X>7+J) - m(^/-1(JBX;7+,) n B^e),

then we can let ô —y 0 (from which e —)> 0 follows) and conclude that

nP(a~1Y G -Bx,7) -^ mo Hjl(Bx^). Since x and 7 were arbitrary the

conclusion then follows since the m oif~^continuity sets Bxr C D\{0}

generate B(D) n D. We first show that nF(a~1Yn G 5^) -> 0. Note

that Zn defined by

Ztnâ sup |/(n,^)| / |dX?|u,ve[o,i] Jo

is a Levy process with jumps bounded by anesupuuGr0-n |/(w,v)| — ane

and that

sup |Yfn| < Z?. (3.20)*e[o,i]

Note that

vz?([anô,oo)) = u^(anB^s) = vy^1(an(B^ô n 50,e)) = 0

since e < ô (^Xn, ^Xn and i/Xi denotes the Levy measures of Zf, X

and Xi respectively, and Bq5,Boj G Hd). Hence, by Theorem 2.3,

nP(a-%n > Ô) -> 0 and hence, by (3.20), nP(fl-1Yn G Bfo) -> 0.

We now show that nF(a'lYn G B^7_s) -> m(HJ1(B^1_5) n 5g>e).First note that

nP(a-1YnG5X;7_,)- nF(a~1Yn G 5x>7-*,Xn g 5(2, ane, [0,1]))

+ nP(a-xYn G 5x>7_Ä,Xn G ß(2,aTCe, [0,1]))

- nP(a-1if/(X") G Bx>7_*,Xn g 5(2, ane, [0,1]))

+ nP(a-xYn G Sx>7_*,Xn G B(2,ane, [0,1]))

- nP(F/(a-1Xn) G Bx>7_*,Xn g B(2,ane, [0,1]))

+ nP(a-xYn G 5X,7_,,XTC G 5(2,aTCe, [0,1]))

(using the fact that Hf is positively homogeneous on V). Note that

nF(Hf(a~1Xn) e B^_s)

= nF(Hf(a~lXn) G Sx,7_j,Xn £ B(2,ane, [0,1]))

+ nP(ff/(a-1Xn) G ßx,7-*,Xn G 5(2, ane, [0,1])).

Since nP(a~1Xn G • ) -^ ra( • H Bq e) applying Theorem 3.7 yields

nF(Hf(a~1Xn) G Sx>7_j) -> m^1^^) n SJ>e).

Moreover,

nPtff/tû-1^) G Sx>7_j,Xn G 5(2,ane,[0,l]))

<nP(Xeß(2,an6,[0,l]))^0,

where the latter convergence follows from Lemma 3.23. Hence

nF(Hf(a-1Xn) G Bx>7_*,Xn g 5(2,ane, [0,1]))

->m(#71(£x>7-*)nBS,e)-

Moreover, by Lemma 3.23,

nF(a-1Yn G Sx>7_*,Xn G 5(2,ane, [0,1]))

< nP(X G 5(2,ane, [0,1])) -> 0.

Hence nP(a;1Yn G 5X,7_5) -> m(Hj1(B^1_ô) n 5£ e). By the same

arguments it follows that nF(a~ Yn G 5Xj7+<s) —y m(HJ (5X;7+j) D


o

I

00 02 04 06 08 1 0

Figure 3.1: 8 simulations of X \ {\XS\ > FT^. ,(0.9) some s G [0,1]},where X is a Levy process with X\ ~ C(l) (a Cauchy distribution with

density fXl(x) = 1/(tt(1 + x2))).

Chapter 4

Dependence in elliptical

distributions

This chapter and the following chapter form the second part of the the¬

sis. In these two chapters we study properties of elliptical distributions.

The class of elliptical distributions provides a rich source of multivari¬

ate distributions which share many of the tractable properties of the

multivariate normal distribution and enables modelling of multivari¬

ate extremes and other forms of nonnormal dependences. The general

representation theorem (Theorem 4.2) allows us to explicitly compute

various interesting quantities and dependence measures without having

to fix a particular elliptical distribution. Moreover, the representation

theorem provides us with a powerful tool for illustrating many different

dependence concepts for nontrivial multivariate models. This chapter is

organized as follows. In Section 4.1 we introduce the class of elliptical

distributions and recall some of its most important properties. We also

prove (Theorem 4.10) that sums of elliptically distributed random vec¬

tors with the same dispersion matrix are elliptical if they are dependent

only through their radial parts. This result has interesting applications

to multivariate time series. In Section 4.2 we study the concordance

measures Kendall's tau (r) and Spearman's rho (qs)- It is easily shown

103

104 Chapter 4. Dependence in elliptical distributions

that for a bivariate normal distributed random vector with linear cor¬

relation coefficient g the relation

2r — — arcsm g,

IT

between Kendall's tau and the linear correlation coefficient, holds. We

show that this relation holds more generally (subject to only slight mod¬

ifications), see Theorem 4.14 below, for all nondegenerate elliptical dis¬

tributions. One prime application of this result is robust estimation

of linear correlation coefficients for nonnormal elliptical distributions.

Monte Carlo studies indicate that this estimator performs better than

most of its competitors (see Figure 4.1 for an illustration). One might

also expect Spearman's rho to be invariant in the class of elliptical dis¬

tributions with continuous marginals and a fixed dispersion matrix. We

give a counterexample showing that this is not true.

4.1 Elliptical distributions

In this section we introduce the class of elliptically distributed random

vectors and give some of their properties. For further details about

elliptical distributions we refer to Fang, Kotz and Ng [17] and Cambanis,

Huang and Simons [11].

Definition 4.1 If X is a d-dimensional random (column) vector and,

for some vector fi G Md, some d x d nonnegative definite symmetric

matrix S and some function (f) : 1R+ —y M, the characteristic function

V?x-M of X —

/i is of the form <^x-M(t) = 0(tTSt); we say that X

has an elliptical distribution with parameters \i, S and (j), and we write

X~Ed(fjL,Z,(f>).

The function 0 is referred to as the characteristic generator of X. When

d = 1, the class of elliptical distributions coincides with the class of one-

dimensional symmetric distributions.

For elliptically distributed random vectors, we have the following general

representation theorem.

4.1. Elliptical distributions 105

Theorem 4.2 X ~ Ed(/J,,T>,(f)) with rank(S) = k if and only if there

exist a random variable R > 0 independent of XJ, a k-dimensional

random vector uniformly distributed on the unit hypersphere §2_ =

{z G Mk | zTz = 1}; and a d x k matrix A with AAT = S; such that

X = /i + RAV. (4.1)

For the proof of Theorem 4.2 and details about the relation between R

and 4>, see Fang, Kotz and Ng [17] or Cambanis, Huang and Simons [11].

Remark 4.3 (a) Note that the representation (4.1) is not unique: if Ö

is an orthogonal k x k matrix, then (4.1) also holds with A' = AÖ and

U' = oTu.

(b) Note that elliptical distributions with different parameters can be

equal: if X ~ F<f(/i, £,0), then X ~ Ed(fi, cS,0c) for every c > 0,

where 4>c(s) = 4>(s/c) for all s > 0.

Example 4.4 Classical examples of elliptical distributions are the mul¬

tivariate normal and the multivariate t-distributions. Let X = \i +

RAXJ ~ F^(/i,S,0), where rank(S) = d. Then X is normally dis¬

tributed if and only if R2 ~ Xd {Xd denotes a Chi Square distribution

with d degrees of freedom), and X is t-distributed with v degrees of free¬

dom if and only if R2/d ~ F(d, u) (F(d, u) denotes an F-distribution

with d and u degrees of freedom).

If the elliptically distributed random vector X has finite second mo¬

ments, then we can always find a representation such that Cov(X) = S.

To see this we use Theorem 4.2 to obtain

Cov(X) = Cov(/z + RAU) = AE(R2) Cov(U) AT,

i.e. Cov(X) exists if and only if K(R2) < oo. To compute Cov(U),let Y ~ Md(0,ld) and let | • |2 denote the Euclidean 2-norm. Then

Y = |Y|2U, where |Y|2 and U are independent. Furthermore |Y|| ^^

Xd, so IE(|"V^12) = d. Since Cov(Y) = Id we see that if U is uni¬

formly distributed on the unit hypersphere in Md, then Cov(U) = Id/d.


Thus Cov(X) = AAT~E(R2)/d. By choosing the characteristic generator

(f)*(s) = (f)(s/c), where c = K(R2)/d, we get Cov(X) = E.

The following result provides the basis of most applications of elliptical

distributions.

Lemma 4.5 LetX ~ Ed(fi, S, (ft), let B be aqxd matrix and letb G Mq.

Then

b + BX~ Eq(b + Bn, BZBT, 0).

Proof. By Theorem 4.2, b + BX has a stochastic representation

b + BX ± b + Bfi + ÜBAU

and the conclusion follows from Definition 4.1. D

If we partition X, \i and S into

where Xi and Hi are r x 1 vectors and Sn is a r x r matrix, then we

have the following consequence of Lemma 4.5.

Corollary 4.6 Let X ~ Ed(ß, S, (p). Then

Xi ~ Er(/ii, Sn, 0), X2 ~ Ed_r(/i2, S22, 0).

Hence, marginal distributions of elliptical distributions are elliptical and

of the same type (with the same characteristic generator).

Next we introduce the linear correlation coefficient for a pair of random

variables with a joint elliptical distribution.

Definition 4.7 Let X ~ Fd(/i,S,0). Fori,j G {l,...,d}, if Y^ > 0

and Tijj > 0, then we call

the linear correlation coefficient for (Xi,Xj)T.

4.1. Elliptical distributions 107

Note that if Var(JQ), Var(X,) G (0,oo), then

Qij = Cov(Xi,Xj)/^Vai(Xl)Vai(Xj),i.e. the linear correlation coefficient as defined by (4.2) is an exten¬

sion of the usual definition in terms of variances and covariances. We

want to interpret the linear correlation coefficient as a scalar measure

of dependence and, as such, it should not rely on finiteness of certain

moments. Clearly (4.2) only makes sense for elliptical distributions. On

the other hand, linear correlation is not always a meaningful measure

of dependence for nonelliptical distributions, whereas Kendall's tau and

Spearman's rho (discussed below) remain meaningful; see for example

Embrechts, McNeil and Straumann [16] p. 25.

A random variable is said to be continuous if its distribution function is

continuous. We now present necessary and sufficient conditions for the

components of an elliptically distributed random vector to be continuous

random variables.

Lemma 4.8 Let X ~ Ed(/i, £,</>), with P(JQ = fii) < 1 for all i G

{1,..., d} and with representation X = fi + RAXJ according to Theorem

4-2. //rank(S) = 1, then Xi,... ,Xd are continuous random variables

if and only if R is continuous. If rank(S) > 2, then Xi,... ,Xd are

continuous random variables if and only ifF(Xi = fii) = 0 for all i, or

equivalently, if and only ifF(R = 0) = 0.

Proof of Lemma 4.8. Let X = fi + RAXJ be a stochastic repre¬

sentation according to Theorem 4.2. Suppose rank(S) = 1, then A

is a d x 1 matrix and U is symmetric {1,—1}-valued. Furthermore,

F(Xi = fii) < 1 implies An ^ 0. Hence, if rank(S) = 1, then Xi,..., Xd

are continuous random variables if and only if R is continuous. Now

suppose rank(S) = k > 2. Define A; = (An,..., A^) and a = AiAj.Since F(Xi = fii) < 1, the case a = 0 is excluded. By choosing an

orthogonal k x k matrix O whose first column is Aj/a and using Re¬

mark 4.3(a) if necessary, we may assume that A^ = (a, 0,..., 0), hence

Xi = fii + aRUi. Note that Ui is a continuous random variable be¬

cause k > 2. Hence F(aRUi = x) = 0 for all x G M\{0}. Hence, if

rank(S) > 2, then Xi,..., Xd are continuous random variables if and

only if F(Xi = fii) = 0 for i = 1,..., d, or equivalently, if and only if

F(R = 0) = 0. D

The following lemma states that linear combinations of independent

elliptically distributed random vectors with the same dispersion matrix

S (up to a positive constant, see Remark 4.3) remain elliptical.

Lemma 4.9 Let X ~ Ed(fi,^,4>) and X ~ Ed(ß, cS,0) for c > 0 be

independent. Then for a,b G M, aX + &X ~ Ed(a/i + bfi, S, (j)*) with

4>*(u) = (j)(a2u)^(b2cu).

Proof. For all t G Md,

= 0((at)TS(at)) 0((&t)T(cS)(6t))= 0(a2tTSt)0(&2ctTSt).

The next theorem shows that this remains true if we allow the elliptically

distributed random vectors to be dependent only through their radial

parts.

Theorem 4.10 Let R and R be nonnegative random variables and let

X = /z + flZ ~ Ed(fi,T1,4)) and X =jl + KL ~ Fd(/I,S,0)? where

(R, R),Z,Z are independent. Then X+X ~ Ed(fi+Ji, S, 0*). Moreover,

if R and R are independent, then 4>*(u) = <fi(u)(f)(u).

For the expression of the characteristic generator, <fi*, we refer to the

proof below.

Proof. Let (f)^r> be the characteristic generator of (R \ R = r)Z, let

4>' be the characteristic generator of Z, and let Fr be the distribution

function of R. Then for all t G

•>oo

cf>'(r2tTXt)ftr\tTXt)dFR(r),

'0

4.2. Kendall's tau and Spearman's rho 109

from which it follows that X + X ~ Ed(fi + fi, S, (j)*), with

<f>*(u) = / (p'(r2u)^r)(u)dFR(r).Jo

Moreover, if R and R are independent, then (p^r\u) = (j)(u) and

(f>*(u) = / (p'(r2u)(j){r)(u)dFR(r)JO

= (f>(u) (p'(r2u)dFR(r) = (f)(u)(p(u).Jo

D

A natural application of Theorem 4.10 is in the context of a multivariate

time series.

Example 4.11 Let Xt = crtTt, t G Z, where the random vectors

Tit ~ Ed(0, S, (pt) are mutually independent and independent of the non-

negative (univariate) random variables at for all t. The cr^'s are allowed

to be dependent. Then for every t G Z, Xt is elliptically distributed

with dispersion matrix S, and so are all partial sums Sn = ^"=1 ^-t- '

4.2 Kendall's tau and Spearman's rho for

elliptical distributions

To begin with we recall the definitions of the concordance measures

Kendall's tau and Spearman's rho. For more on the properties of con¬

cordance measures and in particular on Kendall's tau and Spearman's

rho we refer to Joe [25] and Nelsen [30] and the references therein.

Definition 4.12 Kendall's tau for the random vector (Xi,X2)T is de¬

fined as

t(Xi,X2)±F((Xi-X[)(X2-X'2)>0)-F((X1-X[)(X2-X'2)<0),

where (X[,X2)T is an independent copy of (Xi,X2)T.

Definition 4.13 Spearman's rho for the random vector (Xi,X2)T is

defined as

Qs(Xi,X2) â 3(p((Xi - X[)(X2 - X'i) > 0)

-F((Xi-X[)(X2-X'2')<0)),where (X[,X2)T and (X'{,X2)T are independent copies of (Xi,X2)T.

An important property of Kendall's tau and Spearman's rho is that

they are invariant under strictly increasing transformations of the un¬

derlying random variables. If (Xi,X2)T is a random vector with con¬

tinuous univariate marginal distributions and Ti and T2 are strictly in¬

creasing transformations on the range of X\ and X2 respectively, then

t(Tx(Xi),T2(X2)) = t(Xi,X2). The same property holds for Spear¬

man's rho. Note that this implies that Kendall's tau and Spearman's

rho do not depend on the (marginal) distributions of X\ and X2.

The following theorem relates Kendall's tau and the linear correlation

coefficient for two random variables with a joint elliptical distribution.

Its proof is a combination of Lemmas 4.17 and 4.22 below. For the case

of normal distributions, see also Lemma 4.21.

Theorem 4.14 Let X ~ Ed(fi,^,(p). If for iJ G {l,...,d}, F(Xi =

fii) < 1 and F(Xj = fij) < 1? then

T(Xt,Xj) =(l- ^2(¥(Xi = x))2) 1arcsin^-, (4.3)

^xem. '

where the sum extends over all atoms of the distribution of Xi. If in

addition rank(S) > 2, then (4.3) simplifies to

r(Xi,Xj) = (1 - (F(Xi = fn))2) - arcsin^-, (4.4)7T

which further simplifies to

2r(Xi, Xj) = — arcsin Qi~ (4-5)

7T

ifF(Xi = fil) = 0.

4.2. Kendall's tau and Spearman's rho 111

Example 4.15 Let X ~ Ed(fi,T,,(j)), where rank(E) > 2 and F(X{ =

fii) = 0 for every i. Let H denote the distribution function of X and

let Fi denote the distribution function of Xi. If Fi,..., Fd are arbitrary

continuous distribution functions and for every i, F~l(u) = inî{x G M :

Fi(x) > u} with the convention inf 0 = oo, then

XA(F-1(Fi(X1)),...,F-\Fd(Xd)))T

has univariate marginals Fi,...,Fd and r(Xi, Xj) = r(Xi,Xj). In par¬

ticular, gij = sin(7TT(Xi, Xj)/2) which means that Theorem 4.14 en¬

ables parameterization of multivariate models constructed by marginal

transformations of elliptical distributions.

As a consequence of Theorem 4.14 we have the following well-known

result for Spearman's rho, for which we give an easy proof for complete¬

ness.

Corollary 4.16 LetX ~ Afd(fi, E); where fori, j G {1,..., d}, E^ > 0;

Hjj > 0. Then

gs(Xi,Xj) = - arcsin(gij/2). (4.6)7T

Proof. Recall that g^ = E^-/^/E^E^-. Let Xi = Xi for i = 1,..., d

be mutually independent, and independent of X. Then X ~ Nd(fi, E),where E = diag(En,..., Edd). Hence, X* = X-X ~ Afd(0, E*), where

E* = S + E. Let q\. â E*./V/Ë5Ë*-. Then,

gs(Xi,Xj) = 3t(X*,X*) =31— arcsinp^ J = - arcsin(^/2),

where the second equality follows from Theorem 4.14 and the fact that

the dispersion matrix of a sum of two independent identically distributed

elliptical random vectors differs from those of the terms by at most a

positive constant factor. D

In the light of Theorem 4.14 one might expect Spearman's rho to be in¬

variant in the class of elliptical distributions with continuous univariate

marginals and a fixed dispersion matrix. However, the counterexample

below shows this to be not true.

Counterexample. Let X ~ M2(fi, E), where En, E22 > 0. According

to Theorem 4.2, X has a stochastic representation X = fi-\-RAXJ, where

R ~ xi- We construct a counterexample by deriving a relation between

Spearman's rho and the linear correlation coefficient for the bivariate

elliptically distributed random vector W = AXJ. The relation is given

by

,rrr rrr.

„

/arcsino\t

/arcsino\es(wi.wâ) = 3(—^J-4(—r-^J ,

where g = Ei2/v/^ii^22- For a proof, see Section 4.4. This relation

differs from the relation (4.6) between Spearman's rho and the linear

correlation coefficient for a bivariate normal distribution. The differ¬

ence gs(Xi,X2) — gs(Wi,W2) as a function of the linear correlation

coefficient g is plotted in Figure 4.2. It should be noted that there

are other choices of R (other than R? ~ xi) ^OI which the difference

gs(Xi,X2) — gs(Wi, W2) becomes much bigger.

4.3 Proof of Theorem 4.14

The following lemma gives the relation between Kendall's tau and the

linear correlation coefficient for elliptical random vectors of pairwise

comonotonic or countermonotonic components. It proves Theorem 4.14

for the case rank(E) = 1.

Lemma 4.17 Let X ~ Ed(fi, E, (p) with rank(E) = 1. If F(Xi = fii) <

1, and F(Xj = fij) < 1, then

r(X,,Xj) =(l- ^2(¥(Xi = x))2) 1 arcsin^-. (4.7)^

xem.'

Proof. Let X be an independent copy of X. Let X = fi + RAU and

X = fi + RAU be stochastic representations according to Theorem 4.2,

4.3. Proof of Theorem 4.14 113

where (R, Ü) denotes an independent copy of (R, U). In particular, A

is a d x 1 matrix and U is symmetric {1,—1}-valued. Furthermore,

F(Xi = fn) < 1 and F(Xj = fij) < 1 imply An / 0 and AjX / 0.

Therefore,

Qij = AiiAjl/^JA\A2X = sign^i^ji) = - arcsin^-, (4.8)

(Xi - Xi)(Xj - Xj) = AitAji(RU - RÜ)2 and

F(RU = RÜ) = ^2(F(RU = x))2 = ^2(F(Xi = x))2. (4.9)

If AnAji > 0, then by Definition 4.12

r(Xi,Xj) = F((RU - RÜ)2 > 0) = 1 - F(RU = RÜ)

Using (4.8) and (4.9), the result (4.7) follows. If AiïAjl < 0, then

r(Xi,Xj) = -F((RU - RÜ)2 > 0)

and the result (4.7) follows in the same way. D

Lemma 4.18 Let X ~ Ed(fi, E,0) with rank(E) = k > 2 and let X

be an independent copy of X. If F(Xi = fii) < 1, then F(Xi = Xi) =

(F(Xt = fn))2.

Proof. Let X = fi + RAXJ be a stochastic representation according

to Theorem 4.2. Define A; = (An,... ,A^) and a = AiAj. Since

F(Xi = fii) < 1, the case a = 0 is excluded. By choosing an orthogonal

k x k matrix Ö whose first column is Aj/a and using Remark 4.3(a) if

necessary, we may assume that A^ = (a, 0,..., 0), hence Xi = fii+aRUi.Note that Ui is a continuous random variable because k > 2. Hence

F(aRUi =x)=0 for all x G M\{0}, and it follows that

F(Xi = Xi) = ^2(¥(Xi = x))2 = £)(P(aÄ*7i = x))2 = (F(Xi = fit))2xei set

Lemma 4.19 Let X ~ Ed(fi, E,0) with rank(E) = k > 2, and let X

be an independent copy of X. If F(Xi = fii) < 1 and F(Xj = fij) < 1,

then

r(Xt,Xj) = 2P((X, - Xi)(Xj - Xj) > 0) - 1 + (F(Xt = fn))2. (4.10)

Proof. Since Y = X — X ~ Fd(0,E,^>2), there exists a stochastic

representation Y = RAXJ according to Theorem 4.2. By Lemma 4.18,

F(Yi = 0) = (F(Xi = fn))2 < 1 and similarly F(Yj = 0) < 1. Define

Ai = (An,... ,Aik) and Aj = (Aji,... ,Ajk). With the same argu¬

ments as in the proof of Lemma 4.18, it follows that A;U and AjXJ are

continuous random variables, which implies that P(A;U = 0) = 0 and

P(AjU = 0) = 0. Therefore,

F(YiYj = 0) = P(Ä = 0) = F(Y, = 0) = (¥(Xi = fi^)2.

Since r(Xi,Xj) = 2¥(YiYj > 0) - 1 + F(YiYj = 0), the conclusion

follows. D

Lemma 4.20 Let X ~ Ed(0, E,0) and X ~ Fd(O,cE,0) with c > 0

and rank(E) >2. If ¥(Xt = 0) < 1 and F(X{ = 0) < 1, then

F(XiXj > 0)(1 - F(Xt = 0)) = ¥(XiXj > 0)(1 - P(Xj = 0)).

Proof. Take X = RAXJ according to Theorem 4.2 and set W = AU.

Then

F(XiXj > 0) = F(RWiRWj > 0)

= F(RWiRWj > 0 | R > 0) F(R > 0)

= P(WiWi > 0)F(R>0).

Furthermore, X = a/cäW according to Theorem 4.14, and a similar

calculation shows

F(XiXj > 0) = F(cR2WiWj > 01R > 0) ¥(R > 0)

= P(^W7- >0)P(A>0).

As in the proof of Lemma 4.18, it follows that Wi has a continuous

distribution. Therefore, ¥(R > 0) = 1 - ¥(Xt = 0) and ¥(R > 0) =

1 - F(Xi = 0), and Lemma 4.20 follows. D

Although the next result for normal distributions is well known, we give

a proof for completeness of the exposition and for showing where the

arcsin comes from.

4.3. Proof of Theorem 4.14 115

Lemma 4.21 Let X ~ JVd(/z,E). // ¥(X{ = fii) < 1 and F(Xj

fij) < 1, then

2

r(Xi,Xj) = 2F((Xi - Xi)(Xj - Xj) > 0) - 1 = - arcsin^-,7T

where X is an independent copy of X.

Proof. Using ai = \/E^ > 0, Oj = sj T>jj > 0 and Qij = Ejj/cr^crj, we

have

ynj _ [^H ^U ] _ [ Gi Vi^jQij \

Define Y = X - X and note that (Yi, Yj) ~ Af2(0, 2Hij). Furthermore,

(Yi,Yj) = \l/2(ö-iVcos^j + aiW sin (fij, ajW), where ifij = arcsin^j G

[—7r/2, 7t/2] and (V, W) is standard normally distributed. By the radial

symmetry of (Yi, Yj),

r(Xi,Xj) = 2P(riyi>o)-i= 4P(F; >0, Yj >0)-l

= 4F(V cos <fij+ W sin (fij >0,W >0)-l.

If $ is uniformly distributed on [—ir, tt), independent ofR= \/V2-\- W2,then (V, W) = Ä(cos $, sin$) and

r(Xi,Xj) = 4 P(cos $ cos ifij + sin $ sin (fij > 0, sin $ > 0) - 1

= 4 P($ G ((fij - tt/2, ifij + tt/2) n (0, tt)) - 1

ifjj + tt/22tt

which simplifies to (2/7r) arcsin ^-. D

Lemma 4.22 Let X ~ Ed(fi,Z,(p) with rank(E) = k > 2. If F(X{ =

fii) < 1 and F(Xj = fij) < 1, then

r(Xi,Xj) = (1 - (F(Xi = fn))2) 1 arcsin^-. (4.11)

Proof. Let X be an independent copy of X. By Lemma 4.19, we can

use (4.10). By Lemmas 4.9 and 4.18, X-X ~ £d(0, E, 0*) withPpQ =

Xi) = (F(Xi = fii))2 < 1 and F(Xj = Xj) = (F(Xj = fij))2 < 1. If Z,

Z ~ Nd(fi, ^]/2) are independent, then Z — Z ~ A/d(0, E). By Lemma

4.20,

F((Xi-Xi)(Xj-Xj) > 0) = ¥{{Zi-Zi){Zj-Zj) > 0)(l-(F(Xi = fn))2).

Substituting this into (4.10) and using Lemma 4.21, the result (4.11)follows. D

4.4 Proof of the counterexample

In this section we give a more detailed version of the counterexample

already discussed in Section 4.2.

Counterexample. Let X = fi+RAXJ ~ E2(fi, E, cp), where En, E22 >

0 and fi, R, A and U are as in Theorem 4.2. To construct a counterexam¬

ple we derive the relation between Spearman's rho and the linear corre¬

lation coefficient g = Ei2/VEiiE22 for W = AU. We only consider the

case with rank(E) = 2, since the case with rank(E) = 1 is trivial. From

the invariance of Spearman's rho under componentwise strictly increas¬

ing transformations of the underlying random vector we can without

loss of generality assume that En = E22 = 1 and Ei2 = E2i = g. We

show that the following relation holds,

,rrr rrr.

„

/arcsino\t/arcshifA

., _,

Qs(Wi,W2) =

3{—^-j -4(—^-j (4.12)

In the case of a bivariate normal distribution, i.e. R ~ x%-> we know

from Corollary 4.16 that the relation between Spearman's rho and the

linear correlation coefficient is

Qs(XuX2) = -MC8m(Q/2). (4.13)7T

Since these two relations differ (the difference is plotted in Figure 4.2)we conclude that, contrary to Kendall's tau, Spearman's rho is not

4.4. Proof of the counterexample 117

invariant in the class of elliptical distributions with a fixed dispersion

matrix. It remains to be shown that (4.12) holds. This can be done

following the steps below.

Step 1. Let (Wi,W2), (W[, W£) and (W[', Wl[) be independent copies.

Then

gs(Wi,W2) = 12F(W[ < Wi,W2 < W2) - 3.

Step 2. For (Wx, W2), W[, W2' as above we have that,

F(W[ < Wi,W2 < W2)

1 f27r fl 1. , . ,

= — / —I—arcsm(sin arcsmg + t )27tJq \2 TT

V V "

1arccos cost

2ttv ;

r arccos(cost) arcsin(sin(arcsin g + t)) ) dt.7TZ J

Step 3. The following equalities hold:

/>2tt

(i) / arcsin(sin(arcsin g + t))dt = 0.Jo

/»2tt

(ii) / arccos(cosi)dt = 7T2.Jo

(iii) / arccos(cos t) arcsin(sin(arcsin g + t))dtJo

2t• ^3

^•

= -( arcsm g) arcsm g.

3VH)

2H

Combining Steps 1-3 yields (4.12),

gs(Wi,W2) = 12F(W[ < Wi,W2' < W2) - 3

12 / tt 2, . xo

1.

\0

= — 7T tt (arcsmg) -\— arcsm g — 3

2tt V 2 3tt2v *'

2 V

/arcsinö\ /arcsino\=

\-^r)-\-ir)

Proof of Step 1. Straightforward computations of Spearman's rho

for continuous random variables yields

Qs(Wi,W2) = 3(2F((Wi-W[)(W2-W%)>0)-1)

= 3 (4F(W[ < Wi,W2 < W2) - 1)

= 12F(W[ < Wi,W2 < W2)-3.

D

Proof of Step 2. Let (p,(p',(p" ~ U(0,27r) be independent. Then

(Wi,^) = (cos <£?, sin(arcsin £> + ip)),

(W[,W2) = (cos^/,sin(arcsin^ + (^/)),

(W", W2) = (cos if", sin(arcsin g + f")).

Since

F(W[ < Wi,W2' < W2)

= P(cos if' < cos if, sin(arcsin g + if") < sin(arcsin g + if)),

conditioning on if yields,

F(W[ < Wi,W2 < W2)

i r2ir— / P(C0S t - COS if' > 0)2ttJo

2tt

2î~wu

P(sin(arcsin g + t) — sin(arcsin g + if") > 0)dt.

The factors in the integrand can be written as

P(cosi — cos if' > 0) = 1 2arccos(cost)27T

and

P(sin(arcsin g + t) — sin(arcsin g + if") > 0)

= 1 — — (tt — arcsin(sin(arcsin g + t)) — arcsin(sin(arcsin g + t)))

= —| 2 arcsin(sin(arcsin g + t)).2 27T


Combining these expressions yields,

P(cost — cos f' > 0) P(sin(arcsin g + t) — sin(arcsin g + if") > 0)

= - H— arcsin(sin(arcsin g + t)) — —— arccos(cost)2 7T 27T

arccos(cost) arcsin(sin(arcsin g + t)).

D

Proof of Step 3. (i) and (ii) are elementary. To compute (iii) we

first split the integral depending on arccos(cost) and then use a variable

transformation to obtain

1 f2n

1 =— / arccos(cos t) arcsin(sin(arcsin g + t))dtTT2,KJO

i ( r=—- / t aicsm(sin(avcsin g + t))dt

71-2 \Jo

f2n \+ / (27T — t) arcsin(sin(arcsin £> + t))dt J

1 / A'^'+a'rcsin g

(u — arcsin g) arcsin(sinu)duTT2

,

,"l

-' arcsin g

/»27r+arcsin g

+ / (27T — u + arcsin g) arcsin(sinu)duJ 7r+arcsin g


Hence,

/

7TZ/ (u — arcsin g) arcsin(sin u)du/o

V—; ;'

/»arcsin g

- (u — arcsin g)uduJo

II

/»7r+arcsin g

+ (u — arcsin g)(it — u)dun

s.

III

/»2tt

+ / (27T — u + arcsin g) arcsin(smw)dwJ n

IV

/»7r+arcsin g

/ (27T — u + arcsin g) (w — u)dun

V

/»27r+arcsin^

+ / (27T — u +arcsin g)(u — 27t) duJl-Ks

v 7VI '

The different parts can now be computed separately.

fTÏ /»7T/2

1=1 (u — arcsin g) arcsin(sinu)du = / (u — arcsin g)uduJo Jo

+ I (u — arcsin g)(it — u)du = 7r3/8 — 7r2 (arcsin g)/AJ-k/2

/»arcsin g

II = / (u — arcsin g)udu = — (arcsin g)3/QJo/»7r+arcsin g

III = / (u — arcsin g)(n — u)duJ TV

= — 7r(arcsin g)2/2 + (arcsin g)3/6


/»Z7T

IV = / (27T — u + arcsin g) arcsin(sinu)duJ n

/»3?r/2= / (27T — u + arcsin g) (it — u)du

J n

/>2tt

+ / (27T — u + arcsin g) (u — 2-7r)dwJz-k/2

= - 7r3/8 - 7T2(arcsin g)/A/»7r+arcsin g

V = / (27T — u + arcsin g) (it — u)duTV

= — 7T (arcsin g)2/2 — (arcsin g)3/G/»27r+arcsin g

VI = / (27T — u + arcsin g) (u — 2it)du = (arcsin g)3/Q

Putting everything together yields

1 = \(I-II + III + IV -V + VI)7TZ

\ (1,. sQ

TT2

TT"2" V3=

—g ( - (arcsin £>)3 —— arcsin £>

2, . ^ 1(arcsin g) — - arcsin g.

3tt2Vu/

2

D


Standard Estimator

0 500 1000 1500 2000 2500 3000

Kendall's tau Transform

0 500 1000 1500 2000 2500 3000

Figure 4.1: Linear correlation estimates for 3000 independent samples

of size 90 from a bivariate t$-distribution with linear correlation coeffi¬

cient 0.5. The lower figure shows linear correlation estimates using the

estimator sin(7rr/2) where r denotes the Kendall's tau estimator.


0.003

0.002

0.001

Figure 4.2: The difference between (6/7r) arcsin(^>/2) and

(3/V) arcsin g— (4/7T3) (arcsin £>)3 as a function of g for g G [0,1]

(see the counterexample in Section 4-2).

Chapter 5

Multivariate extremes

for elliptical distributions

In this chapter we analyze multivariate extremes for elliptical distri¬

butions. The analysis also highlights various aspects of the concept

of multivariate regular variation which was introduced in Chapter 1.

Recall from Theorem 4.2 in the previous chapter that any elliptically

distributed random vector has a stochastic representation of the form

(see Theorem 4.2 for details)

X = fi + RAV, (5.1)

where the nonnegative random variable R and the random vector U are

independent. This simple structure of elliptical distributions enables

explicit computations of interesting quantities such as the coefficients

of tail dependence (see Definition 5.1 below) and spectral measures as¬

sociated with regularly varying random vectors (see Theorem 1.15).

This chapter is organized as follows. In Section 5.1 we prove that for

an elliptically distributed random vector X with representation (5.1),X is (multivariate) regularly varying with index a > 0 if and only if

R is regularly varying with index a > 0. We also show that if R is

125

126 Chapter 5. Extremes for elliptical distributions

regularly varying with index a > 0 and Qij > — 1 (see Definition 4.7),then (Xi, Xj) has tail dependence (see Definition 5.1 below) and we de¬

rive an expression for the coefficient of tail dependence from which we

conclude that the coefficient of tail dependence is fully determined by

the corresponding linear correlation coefficient (as defined in Definition

4.7) and the tail index of the radial random variable R in the general

representation. In Section 5.2 we explicitly compute the spectral mea¬

sure associated with a regularly varying elliptically distributed random

vector with respect to the usual Euclidean norm and the max-norm,

respectively. We find that the spectral measure depends only on the

choice of norm, the tail index a and the dispersion matrix S. Moreover,

the explicit expressions for the spectral measures allow us to interpret

the effect of the choice of norm on the spectral measure but also to

illustrate the multivariate tail behavior of regularly varying elliptically

distributed random vectors.

5.1 The connection between regular varia¬

tion and tail dependence

Perhaps the most commonly encountered measure of dependence of bi¬

variate extremes is the coefficient of upper (lower) tail dependence.

Recall that for a univariate distribution function F we denote by F_1

its (left-continuous) generalized inverse given by F~l(u) = inf{x G M :

F(x) > u] with the convention inf 0 = oo.

Definition 5.1 Let (Xi,X2)T be a random vector with marginal dis¬

tribution functions F\ and F2. The coefficient of upper tail dependence

of (Xi,X2)T is defined as

\u(Xi,X2) 4 limP(X2 > Fï\u) \ Xx > F^(u)),u-fl

provided that the limit \u(Xi,X2) G [0,1] exists. The coefficient of

lower tail dependence is defined as

Xl(Xi,X2) 4 limP(X2 < F~l(u) | Xi < F^(u)),

5.1. Regular variation and tail dependence 127

provided that the limit Xl(Xi,X2) G [0,1] exists. If \u(Xi,X2) > 0

(Xl(Xi,X2) > 0), then we say that (Xi,X2)T has upper (lower) tail

dependence.

Elliptically distributed random vectors are radially symmetric. Hence,

if (Xi,X2)T is elliptically distributed, then Xu(Xi,X2) = Xl(Xi,X2).If XU(X1,X2) = XL(Xi,X2) > 0, then we say that (Xi,X2)T has tail

dependence. See e.g. Nelsen [30] for details on radial symmetry. Note

that the tail dependence coefficients need not exist, see e.g. Example

1.24.

For a pair of random variables, upper (lower) tail dependence is a mea¬

sure of joint extremes. That is, it measures the probability that one

component is extremely large (small) given that the other one is ex¬

tremely large (small), relative to the marginal distributions. We have

introduced two concepts for measuring dependence of multivariate ex¬

tremes of random vectors, the coefficient of tail dependence (Definition

5.1) and the spectral measure associated with a regularly varying ran¬

dom vector (Definition 1.17). In the next theorem we clarify the connec¬

tion between these two concepts. We also derive an explicit expression

for the coefficient of tail dependence for two random variables with a

joint elliptical distribution.

Theorem 5.2 Let X = /i + RAXJ ~ Ed(fi,Z,(p), with S^ > 0 for

i = 1,..., d, and where fi, R, A and XJ are as in Theorem 4-2. Then

the following statements hold.

(i) R is regularly varying with index a > 0 if and only ifX is regularly

varying with index a > 0.

(ii) If R is regularly varying with index a; > 0, then

-tt/2

XU(X1,X3) = XL(X,,X3) =J{v/2-"eii)/*~" "". (5.2)f

'cosatdt

J (n/2—arcsin gij)/2

f*/2 cos« tdt

Remark 5.3 By applying Theorem 2.1 in Bingham and Inoue [8] one

can show that if £^ > 0, T,jj > 0, \gij\ < 1 and Xu(Xi,Xj) > 0 (or

equivalently Xl(X1,X3) > 0), then R is regularly varying with index

\ogF(R>u)a = — lim sup

u^oo logW

if, for some e > 0 and some a G (0,1), a is the only zero of

r1 uan

r uz,

r uan

r1 uz.du / du — / du / du

'o Vi — u2 Jo y/1 — u2 Jo \/l — u2 Jo y/ï u2

in {w G C : Ke(w) G (—e + a, e + a)}. Computations (using the software

package Maple) for a large number of a indicate that this is likely to

hold (i.e. that Xu(Xl,X3) > 0 implies that R is regularly varying).

Remark 5.4 Note that the tail dependence coefficient (5.2) is increas¬

ing in gl3 and decreasing in a. Also note that

C/2 cos" tdt.. J(tt/2—arcsin ol7)/2 _

hm ——

tt—— = 0.a^°° f*' cos« tdt

Remark 5.5 Let X ~ Ed(fi,T,,(p) with Xz ~ F% and X3 ~ F3. Note

that if limw|i F~ (u) < oo, i.e. if X% is a bounded random variable,

then there exists a uq G (0,1) such that the events {Xt > F~x(u)} and

{X3 > F~l(u)} are disjoint for u > uq, and hence

F(Xl>F-1(u),X3>F-1(u))hm i

= 0,«ti F(Xl>F~1(u))

i.e.Xu(Xl,X3) = XL(Xl,X3) = 0.

From the theorem above we can conclude that the bivariate marginals

of an elliptically distributed vector X have tail dependence if the radial

random variable R in the representation X = fi + RAXJ is regularly

varying with index a > 0. The linear correlation coefficient gl3 only ef¬

fects the magnitude of the coefficient of tail dependence. An interesting

consequence is that if X ~ Ed(fi, S, (p), then (Xt, X3)T can have a coef¬

ficient of tail dependence significantly larger than zero even if the linear

5.1. Regular variation and tail dependence 129

correlation coefficient for (Xi,Xj)T is zero or negative. In Figure 5.1

we have plotted the coefficient of tail dependence for a regularly vary¬

ing elliptically distributed bivariate random vector with uncorrelated

components as a function of the tail index a.

Proof of Theorem 5.2. (i) By Corollary 1.33 we can without loss

of generality assume that fi = 0. If rank(S) = k < d, denote by£(_1) A (A(-1))tA(-1) the generalized inverse of £, where A^-1) =

(ATA)~1AT, i.e. A(_1) solves A^~Â = Ik, where Ik denotes the k x k

identity matrix. Note that £(_1) = S_1 if rank(E) = d. By choosing

the norm |x|E = (xtTj^~1>x)1'2 in the definition of regular variation (by

Corollary 1.20 we are allowed to choose any norm), we obtain

P(|X|E > tx, X/|X|S G • ) ¥{R > tx, AXJ e )

|X|S > t) ¥(R > t)

F(R > tx) F(AU G • )

¥(R > t)

,—aIf R is regularly varying, then limôo F(R > tx)/F(R > t) = x~

and hence P(|X|S > ta,X/|X|E G • )/P(|X|E > t) A- x~aF(AV G • )as t ^ oo. Conversely, if P(|X|E > ta,X/|X|E G • )/P(|X|E > t) -^

x-a p^Q ^ . ^ as ^ _^ 00^ then we must have O = AXJ and limôo F(R >

tx)/F(R>t) =x~a.

(ii) For Qij e {—1,1} the conclusion follows immediately. Therefore we

only consider the case \gij\ < 1. First note that if Xu (Xi — fii, Xj —

fij )

exists, then Xjj(Xi — fii,Xj —fij) = Xu(Xi:Xj). Hence, we can without

loss of generality take fi = 0. Note that if X ~ Ed(0, S, (p), with Xi ~ Fi

and Xj ~ Fj, then F~l(u) = x/Ë~fË~F~1(u) for u G (0,1). Note also

that if limw|iF~1(u) < oo, i.e. if Xi is a bounded random variable,

then there exists a uq G (0,1) such that the events {Xi > F~x(u)} and

{Xj > F~l(u)} are disjoint for u > uq, and hence

F(Xl>F-1(u),X3>F-1(u))hm i

= 0.«ti F(Xi>Fl-1(u))

Hence, if Xu(Xi,X3) G (0,1] exists, then

F(Xi>^fY~iZ,Xj > y/Ë^z)Xv(Xi,Xj) = lim

F(Xi > ^JT~iz)

Since X = RAXJ,

'Xi \ d ( y/â 0\ ( cos f

R

T

\xjJ \ y/ZjjQij y/^jjyi-ßij ) \sin^

where if ~ U(—tt,tt), i.e.

(Xi,Xj)T = (y/Ë~iR cos (f, x/^J](gijR cos if + yj1-g2jR sin if))

= (y/ûRcosif, \/ÊjjRsin(aicsingij + (f))T.

Hence, if Xjj(Xi,X3) G (0,1] exists, then

, , F(R cos (f > z,Rsin(axcsin gij + if) > z)XjjiXi. Xj)

= hm ——

r .UK *' JJzôo F(Rcosf>z)

The numerator can be written as

F(Rcos if > z, R sin(arcsin gi3 + if) > z)1 1

= F(R> zmax(,

-—

: r),vcos if sin (arcsin g^ + if)

cos if > 0, sin(arcsin g^ + ip) > 0)

= - / P(i2> z/cost)dt,** J (tt/2—arcsin gij)/2

and the denominator can be written as

P(Ä cos 99 > z) = F(R > z/ cos if, cos 99 > 0)

1 W2

= - / F(R> z/cost)dt.

Suppose there is an a > 0 such that for every x > 0

¥(R > zx)j F(R > z) -r> x~a as z -+ 00, (5.3)

i.e. suppose Ä is regularly varying with a > 0. By Theorem 1.5.2 p. 22

in Bingham, Goldie and Teugels [7], for every 0 < a zx)j F(R > z) -r> x~a as 2 - 00,

5.2. Explicit computations of spectral measures 131

uniformly in x on [a, b]. In particular, with x = 1/ cost,

F(R> z/cost)/F(R> z) ^cosat as z -)> oo,

uniformly in £ on [0,7r/2 — e] for every e G (0,7r/2) Hence, for every

e G (0, tt/2) and a G [0, tt/2 - e],

r/2F(R> z/cost)n

r/2~ea ,

hmsup /m/n

'—-—-dt < / cosatdt + e

z^JJa F(R>Z) -Ja

n

r/2F(R>z/cost)n

r/2~e„ ,

liminf/v

, „

'

,

'dt > / cos" Mt.

p ä > z-

A0—»-OO I

If we let

C'%.

./0¥(R> z/cost)dtw \ A J(7r/2-arcsmgi3)/2 V / /

f*/2F(R> z/cost)dt

then

f^/2-6 cos« tdtJ(ff/2-arcsinglJ)/2

< liminfA(z)J^7 ecos«tdt + e

z^°°

< lim sup A(z)z—>oo

<

r/2~e cos" tdt + eJ (tt/2—arcsin gl3 )/2

S;/2~e cos« Mt

Letting e —>• 0 yields

f71-/2 cos« ^\ /v V \

—

J(^/2-arcsineîJ)/2

/0X cos« tdt

Moreover, because elliptically distributed random vectors are radially

symmetric about fi, X\j(Xi,X3) = Xi,(Xi,X3). D

5.2 Interpretations and explicit computa¬

tions of spectral measures

In this section we discuss how to interpret the spectral measure with

respect to different norms. The discussion is general but in the case of

elliptical distributions we can explicitly compute the spectral measure

with respect to different norms and compare different choices. Real

data, e.g. financial asset price log returns, often indicate that the un¬

derlying distribution is elliptical or at least close to elliptical, and many

statistical models are based on the assumption of ellipticality. Hence

the following discussion should be relevant for many applications, espe¬

cially in risk management. See Breymann, Dias and Embrechts [10] for

an interesting empirical study of dependence and extremes for bivariate

time series of foreign exchange data.

By Corollary 1.20 we know that if a random vector X is regularly varying

with respect to some norm on IRd, then it is regularly varying with

respect to every norm on FLd. For every choice of the norm the spectral

measure is a measure of dependence between extreme values. However,

the choice of norm becomes essential when interpreting the spectral

measure. The choice of norm must be related to the question we are

trying to answer. A natural question would be: What is the dependence

between the components of a random vector given that at least one of

its components is extreme? In the literature (see e.g. Stäricä [39]) most

authors consider the Euclidean 2-norm, | • |2- However, if we want a

measure of dependence between the components - given that at least

one of the components is extreme - then we should use the max-norm

IXloo = max{|Xi|,..., |-Xd|}. Clearly, if we take x = 1 in equation (1.2)of Theorem 1.15, we have that

P(eoo G • ) 4 hm P(X/|X|00 G • | ixi«, > t)t—>-oo

= hmP(X/|X|00G-md=1{|Xj|>t})

from which it is seen that the max-norm corresponds to the question

posed. However, if the components are not identically distributed,

then it might be more natural to condition on the event Uj=1{|Xj| >

G~l(u)}, where Gj is the distribution function of \Xj\ and u t 1- For

X ~ Ed(0, S, <p) this is achieved by considering the weighted max-norm

Ix|oo,e = max{|Xi|/v/Ëïï, •••> \xd\/y/^dd}, since in this case,

P(eoo,E G • ) = lim P(X/|X|00;E G • I ixi^e > t)

= limP(X/|X|00;E G • | Ud=i{\X3\ > G-\u)}),


is the spectral measure of X with respect to the norm | • |oojE. The

corresponding question in this case would be: What is the dependence

between the components of a random vector given that at least one of

its components is extreme relative to its marginal distribution?

In the following two examples we compute the spectral measure with

respect to the Euclidean 2-norm and the max-norm for bivariate regu¬

larly varying elliptical distributions. This can also be done for elliptical

distributions of higher dimension, but the corresponding computations

in spherical coordinates become quite tedious.

Example 5.6 Let X ~ E2(0,T,,(p), with En, £22 > 0> t>e regularly

varying with index a > 0, and let X = RAXJ be a stochastic repre¬

sentation according to Theorem 4.2. Without loss of generality we can

choose A and U such that

/Xi\ ±R( V^Ti 0 \ /cosyA

\X2) y yf^22g\2 y/T^/i^/l - q\2 J \simf)'

where p ~ U(—ir/2, Sir/2), i.e.

(Xi,X2)T = (y/ËïÏRcosip, yfË2~2~(gi2Rcosif + y 1 - g22Rsimf))T

(y/EiiRcosif, y/^22Rsin(arcsing12 + f))T

Let

f(t) = (Sn cos2 t + E22 sin2(arcsin gi3 +t)) ,

f -tt/2, t = -7r/2,

g(t) 4 J arctan (^§^12 + V1 ~ &tant)) ,t G (-tt/2, tt/2),

{ g(t-7r) + 7T, te [7r/2,37r/2).

Then,

RAv=R\Avhj§r±Rfacosf):\AXJ\2 \smg(if)/

Since X is regularly varying and X/|X|2 has continuous distribution on

S2, there exists a random vector O such that for every x > 0 and every

S B(S£),

lim m\AVh>rAV/\AVh e 5)=

z^oo ¥(R\AXJ\2 > z)K '

Moreover, by Theorem 5.2, R is regularly varying which implies that

there exists a slowly varying function L such that F(R > x) = x~aL(x).Let Sgljg2 = {(cost, sint)T : t G (0i,02)}, where by symmetry we can

assume that —tt/2 < 9i < 92 < tt/2. The case \gi2\ = 1 is trivial, so we

consider only the case |^»i2| < 1. Then, for t G (—tt/2, tt/2),

q l(t) = arctan —. ( .tant — gi2

and

F(R\AXJ\2 > zx,AXJ/\AXJ\2 e S)lim

F(R\AXJ\2 > z)

,

J£h$ z-«x-af(trL(zx/f(t))dt= hm ——l-t5

J^z-"f(t)"L(z/f(t))dt

« ,

i;-^f(t)aL(zx/f(t))/L(z)dt—

x hm—-—^—^

z-°° Cf(t)«L(z/f(t))/L(z)dt

J9-u^ f(t)adt

/o /Wad*

-a//-H^O (Sl1 C0S2 t + S22 sin2(arCSm Ö12 + ^))"/2 dt

X

Jo^

(Sn cos21 + £22 sin2(arcsin £12 + t)) dt

The third equality follows from the fact that L(zx)/L(z) —> 1 uniformly

in x on intervals [a, b], 0 < a < b < 00 (Theorem 1.5.2 p. 22 in Bingham,

Goldie and Teugels [7]) and from the fact that there exist constants

0 < ci < c2 < 00 such that c\ < l/f(t) < c2 for all t G [-tt/2,tt/2].Now we can identify the spectral measure as

In-i(fti (Sn cos21 + S22 sin2(arcsin g12 + t))a/2 dt

p(e g s9ue2) =g 2nl} ^72—•JQn (Sn cos21 + £22sin2(arcsin^>i2 + t))" dt

Note that the spectral measure depends on the tail index a. Further¬

more, note that lima^0P(O e S) = F(AXJ/\AXJ\2 e S) for all S G

B(S>2_1). We see that the spectral measure is absolutely continuous and

hence it has a density. The density is plotted in Figure 5.2 for bivariate


regularly varying elliptical distributions with (£n, £22, Q12) = (1,1,0.5)and with tail indices a = 0, 2,4, 8,16 (we write a = 0 for the limit mea¬

sure lima_>oP(0 G • )). From this figure it can be seen that as a

increases - that is as the tails become lighter - the probability mass

becomes more concentrated in the main directions of the ellipse (in this

case 7r/4 and 57r/4). Note also that the density can be explicitly com¬

puted by differentiating expression (5.4).

Example 5.7 Let us now compute the spectral measure with respect

to the norm | • |oo- Proceeding analogously to the previous example but

replacing the function / by

f(t) = max{-\/£ii I cost|, \/£22 | sin(arcsingi2 + t)\},

we find thatAXJ

RAV = RlAXJlov——|-AU loo

where AU/jAU^ = f(if) with f ~ U(-tt/2, 3tt/2). Following the

computations in the previous example we obtain the spectral measure

with respect to the max-norm as

P(© G Sdue2)

//-i(01) (max{^/£77 I cost|, y/T^ | sin(arcsin gX2 + t) | })a dt

JQn (max{\/£ii I cost|, a/^22 | sin(arcsin gi2 +t)\})a dt

where Sq1}q2 is the radial projection of Sq1iq2 (see Example 5.6) on S^,the unit circle with respect to the max-norm. The density of the spectral

measure is plotted in Figure 5.3 for bivariate regularly varying ellipti¬

cal distributions with (£n, £22, £12) = (1,1,0.5) and with tail indices

a = 0, 2,4, 8,16. From this figure it can be seen that as a increases

- that is, as the tails become lighter - the probability mass becomes

less concentrated in the main directions of the ellipse (in this case 7r/4and 57r/4). This is quite intuitive, for (bivariate) regularly varying el¬

liptical distributions with lighter tails, the probability of joint extremes

(that both components are extreme) becomes very small compared to

the probability that one component is extreme. This can be seen from

the fact that the coefficient of tail dependence tends to zero as the tail

index increases (see Remark 5.4 and Figure 5.1).


0.5-

0.4-

0.3-

0.2-

0.1

_- , , , , i , , , , i , , , , i , , , , i , , , , i ,

0 2 4 6 8 10

Figure 5.1: The coefficients of upper and lower tail depen¬

dence for regularly varying bivariate elliptical distributions with

(En, E22, 012) = (1,1, 0), as a function of the tail index a (see Remark

5.4).

Note the striking difference between the spectral measure with respect to

the Euclidean norm and the spectral measure with respect to the max-

norm. By choosing a norm which does not correspond to the question

one is trying to answer, one might draw completely wrong conclusions

about dependences between extremes. The best illustration of this is

the comparison of Figure 5.2 with Figure 5.3.

! 'n ; ',; i i tit < i,

Figure 5.2: Densities of the spectral measure ofX ~ E2(fi,Yj,(p) with

respect to the Euclidean 2-norm, where (En, £22, 012) = (1,1, 0.5), and

ta«Z index a = 0, 2,4, 8,16. Larger tail indices correspond to higher peaks

(see Example 5.6).

Figure 5.3: Densities of the spectral measure ofX ~ E2(fi,Tj,(p) with

respect to the max-norm, where (En, £22,012) = (1,1,0.5), and tail

index a = 0,2,4, 8,16. Larger tail indices correspond to higher peaks

(see Example 5.7).

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Curriculum Vitae

Personal Data

Name:

Date of Birth:

Nationality:

Education

2001 - date:

1994 - 2000:

1991 - 1994:

Carl Filip Lindskog

20.10.1975

Swedish

Ph.D. student in mathematics at ETH Zürich;

Supervisor: Prof. Dr. Paul Embrechts

Studies in mathematics and physics at the

MSc programme in Engineering Physics,

Royal Institute of Technology (KTH), Stock¬

holm;

MSc awarded in February 2000

Bromma Gymnasium, Stockholm

Employment

2000 - date: Researcher at RiskLab, ETH Zürich

Research Interests

2000 - date: Stochastic processes, multivariate extreme

value theory, dependence modelling, Risk

Management, Mathematical Finance

143

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