the strict domain of attraction of strictly stable law

Japan. J. Math.

Vol. 15, No. 2, 1990

The strict domain of attraction of strictly stable law

with index l

By Takaaki SHIMURA.

(Received October 17, 1989)

•˜ 1. Introduction

The set of strictly stable distributions is a proper subset of the set of

stable distributions. If the index is not 1, then every stable distribution is

a shift of a strictly stable distribution (prescribed in Section 2) . While, if

the index is 1, then there is a stable distribution which is not a shift of a

strictly stable distribution.

A distribution v on Rd is said to be in the strict domain of attraction of

a strictly stable i, if, for some Bn>0, the distribution of Sn=B;1Xk

converges to p, where X1, X2, •c is a sequence of i .i.d. (independent and

identically distributed) random vectors with distribution v . If the index is

not 1, characterization of the strict domain of attraction is essentially

known. However, the case of index 1 is quite different .

The first purpose of this paper is to find necessary and sufficient condi

tions for distributions to belong to the strict domain of attraction of a

strictly stable law with index 1, and the second purpose is to classify

distributions in the strict domain of attraction according to their tail

behavior.

Examples in the last section will illustrate remarkable difference be

tween the case of index 1 and the other cases.•˜

2. Preliminaries

First we introduce some notations. Let Rd be the d-dimensional

Euclidean space. We consider it as the set of column vectors with d com

ponents with inner product xy=J1x,y, for x=(xj)1_??_j_??_d and y(yj)1_??_j_??_d and

with norm |x|=(xx)1/2. Let Sd-1 be the unit sphere in Rd. The o-algebras

of Borel subsets of Rd and of Sd-1 are denoted by _??_(Rd) and _??_(Sd-1),

respectively. The sets of all probability measures on Rd and Sd-1 are

denoted by _??_(Rd) and _??_(Sd-1), respectively. For pe(Rd), the character

istic function of p is denoted by 2. For a random variable X, px denotes

352 TAKAAKI SHIMURA

the distribution of X. For p, vEP1i(Rd), p*v denotes the convolution of p

and v. We call a convolution of p and a delta distribution a shift of p. For

pE~(Rd), pn--~p denotes weak convergence of p to p. For vEP(Rd) and

AE(Rd), xv(dx) denotes xiv(dx)1~j<_d.

DEFINITION 2.1. A distribution pEe,(Rd) is strictly stable if for a>0,

b>0, there exists c>0 such that

(2.1)

It is well known that except in the case of oo (delta distribution at 0)

this c is uniquely determined by a and b, and there exists a with 0<a~2

such that c«=a"+b". This a is called the index (exponent) of the strictly

stable law. Distributions ~a (delta distribution at a) with a•‚0 are called

trivial strictly stable distributions with index 1.

The following theorem characterizes strictly stable laws as limit distri

butions. The proof can be given in a way similar to the case of stable laws.

THEOREM 2.2. A distribution p on Rd is strictly stable if and only if

there exist a sequence X1, X2, •c, Xn, •c of i.i.d. random vectors and positive

numbers Bn such that where

(2.2)

DEFINITION 2.3. Let X1, X2, •c, Xn, •c be a sequence of i.i.d. random

vectors with distribution v. If, for suitably chosen positive constants Bn,

the distribution of (2.2) converges as n-oo to a distribution p, then we say

that v is strictly attracted to p. We call the totality of distributions strictly

attracted to p the strict domain of attraction of p. Instead of (2.2), if, for

suitably chosen constants Bn and AnERd, the distribution of S=B;1Xk

A converges as n--*oo to a distribution p, then we say that v is attracted

to p. As usual we call the totality of distributions attracted to p the domain

of attraction of p. Let D(p) and SD(p) denote the domain of attraction of p

and the strict domain of attraction of p, respectively.

From the above theorem, a distribution has a non-empty strict domain

of attraction if and only if it is strictly stable. It is well known that a dis

tribution has a non-empty domain of attraction if and only if it is stable.

Characterization of the domain of attraction of a stable law is given in B. V.

Gnedenko and A. N. Kolmogorov [3], I. A. Ibragimov and Yu. V. Linnik [4],

W. Feller [2] and E. L. Rvaceva [6].

Strict domain of attraction 353

It is well known that if p is strictly stable with index a, then the normalizing constants Bn in (2.2) is written as

(2.3) Bn=n1/ƒ¿h(n),

where h(x) is a slowly varying function, i.e. h(x) is real-valued , positive and measurable on [A,oo) for some A>0, and, for each k>0, lim n.h(kx)Jh(x)=1([1]).

DEFINITION 2.4. The slowly varying h(x) appearing in (2.3) is called the slowly varying function part of normalizing constant Bn.

DEFINITION 2.5. Distributions P1,P2E1!1(Rd) are strictly type-equivalent if there exists a constant c>0 such that 21(z)=22(cz).

Let p1 and ,u2 be strictly type-equivalent and 21(z)=Mcz). Obviously, their strict domain of attraction are identical. If v is in this strict domain of attraction and if, for i=1,2, Bn ,i is the normalizing constant in (2.2) for pti and hi(x) is the slowly varying function in (2.3), then

(2.4)

If Pi and 2 are not strictly type-equivalent, then SD(pi)nSD(p2)=0. These

facts are proved by the convergence of types lemma ([3], Section 10). Hence

it is possible to speak of the strict domain of attraction of a strict type.

We say that slowly varying functions are equivalent if (2.4) holds for some

c(>0). Thus, with each v in the strict domain of attraction, an equivalence

class of slowly varying functions is associated.•˜

3. Characterization of strict domain of attraction

In this section, we give a necessary and sufficient conditions for distri

butions to belong to the strict domain of attraction of a strictly stable law

with index 1.

A canonical representation of strictly stable laws with index 1 is as

follows. A distribution p on Rd has characteristic function

(3.1)

with reRd, c•L>0, 2eeIJ(Sd-1) satisfying A(d)=0, if and only if p is a

non-trivial strictly stable distribution with index 1.In what follow, we denote x=xE for xeRd\{0}.

354 TAKAAKI SHIMURA

THEOREM 3.1. Let 1 be a strictly stable distribution on Rd with the

characteristic function (3.1). In order that a distribution v on Rd belongs to SD(p), it is necessary and sufficient that the following conditions hold:

(3.2)

for all A-continuous sets E1,

(3.3) for all k>0.

(3.4)

REMARK. The conditions (3.2) and (3.3) together are equivalent to the

condition that for all A-continuous sets F1, F2€(S') and k>0,

(3.5)

If we drop the restriction JeA(d)=0, then (3.1) is the canonical representation of a stable law with index 1 and the domain of attraction of the law in question is characterized by (3.5).

If, in particular, d=1, the above theorem takes the following form.

THEOREM 3.2. Let u be a Cauchy distribution with characteristic function

(3.6) where c>0.

In order that a distribution v on R1 belong to SD(p), it is necessary and sufficient that

(3.7)

(3.8) for all

(3.9)

PROOF OF THEOREM 3.1. By E. L. Rvaceva [6], it is sufficient to show that (3.2)-(3.4) are equivalent to the fact that there exist constants Bn>0 satisfying the following (3.10)-(3.12).

(3.10)

for any A-continuous set E€&(Sd1) and R>0.

(3.11) for


(3.12) for

It can be proved in a similar manner to E. L. Rvaceva [6] that the conditions (3.2) and (3.3) are equivalent to the existence of Bn>4 satisfying

(3.10) and (3.11). Hence, we have to deal only with the last condition. It is easy to see that (3.2)-(3.4) imply (3.12), because we get

using (3.10). Next we prove that (3.10)-(3.12) imply (3.4). From (3.10) and

(3.12),

Hence it is enough to show that

(3.13)

Since we can prove that Bn=max1_??_k_??_n Bk are also normalizing constants

([1], p. 23: monotone equivalents of regularly varying function), we assume that Bn is an increasing sequence. Let

We will prove that limn.V,,= 0. It is noted that

where

Now set

Since

356 TAKAAKI SHIMURA

we see that lim,,~~Vn=0. Here we have used the facts that .R2(Ix!>R) is

slowly varying and that Bn+1/Bn•¨1 as n--goo. By the same reason, we haveand

which imply limVn=O. Therefore limn.V,,=0, which proves (3.13). Hence the proof is complete.

In case a*1, strict domains of attraction are described by the following theorems: Theorems 3.3-3.5.

THEOREM 3.3. Assume that 0<a<1. A distribution p on Rd is strictly stable with index a if and only if

(3.14)

with c•L>0, AEq?i(Sd-1). Let p be strictly stable with index a having charac

teristic function (3.14). Then veSD(p) if and only if v satisfies (3.2) and (3.3)

with k in the right-hand side replaced by ha.

THEOREM 3.4. i) Assume that 1<a<2. A distribution p on Rd is strict

ly stable with index a if and only if

(3.15)

with c•L>0, Aef(Sd-1)

ii) Let p be a distribution with the characteristic function (3.15). Then

vESD(p) if and only if v has mean 0 and satisfies (3.2) and (3.3) with k in the

right-hand side replaced by k".

THEOREM 3.5. A distribution p on Rd is strictly stable with index 2 if

and only if p is Gaussian (possibly degenerate) with mean 0 and p*oo. Let

p be strictly stable with index 2 and let (ajk) be its covariance matrix. Set

A(z)=~,k=1aJkz;zk. Then vESD(p) if and only if it satisfies the following

three conditions:

(3.16)

(3.17)

where z, weRd with A(w)•‚0, and where

Strict domain o f attraction 357

(3.18)

The canonical form of characteristic functions in Theorems 3.1-3.5 are

given in [5] and [8]. The proof of the characterization of the strict domain

of attraction in Theorems 3.3-3.5 can be given in a similar manner to the

proof of Theorem 3.1 if we make use of properties of regularly varying

functions in Theorems 1 and 2 in [2], Chapter 8 Section 9.•˜

4. Classification of the strict domain of attraction

In this section, we show some properties of distributions in the strict

domain of attraction. Having observed the difference between the domain

of attraction and the strict domain of attraction, we have the following two

propositions in the case of a*1. They are consequences of Theorems 3.3-3.5.

PROPOSITION 4.1. Let v be a distribution in the domain of attraction of a

stable law p with index 0<a<I. Then v belongs to the strict domain of

attraction of a strictly stable law ps=p*os with some seRd. More precisely,

if X1, X2, •c is a sequence of i.i.d. random vectors with distribution v and

the distribution of

(4.1)

converges to p, then the distribution of B;1Xk converges to ,us and the

sequence An converges to s.

PROPOSITION 4.2. Let v be a distribution in the domain of attraction of a stable law with index 1<a<2. Then v belongs to the strict domain of attraction of a strictly stable law if and only if v has mean 0.

Propositions 4.1 and 4.2 imply that if v lies in the domain of attraction

of a stable law with index a1, then v itself or a shift of v belongs to the strict domain of attraction of some strictly stable law. However, the case

of index 1 is more delicate and the total behavior of the distribution is relevant.

The main results in this section start with the assertion that any

slowly varying function can appear in the slowly varying function part of the normalizing constants for distributions in the strict domain of attraction of strictly stable distributions. For d=1, H. G. Tucker [10] shows a similar fact for the (non-strict) domain of attraction. But in the case of the strict domain of attraction for index 1 we have to make full use of Theorem 3.1.

358 TAKAAKI SHIMURA

THEOREM 4.3. Let 0<a<2. For any non-trivial strictly stable law p with index a and any slowly varying function h(x), there exists a distribution v on Rd such that p5~---*,cc, where Sn=(n1/"h(n))1h1Xk and Xk is a sequence of i.i.d. random vectors with distribution v.

PROOF. We prove the assertion in the case of a=1 in detail. A distribution v belongs to the domain of attraction of the strictly stable law p with characteristic function (3.1) if and only if there is a slowly varying function H(R) such that

(4.2)

for each A-continuous set

where o(1) is a term which tends to 0 as R--goo. If v satisfies this condition, then B/&/"h(n) is the normalizing constant for v if and only if h(R) and H(R) satisfy the relation

(4.3)

We are now in a position to construct such a distribution u and a function H(R) satisfying (4.2), (4.3) and (3.4) for given h(x). It follows from the existence theorem of asymptotic inverses of regularly varying functions ([7],

p. 23) that, for each given h(x), there exists a slowly varying function H(R) satisfying (4.3). By the representation theorem of slowly varying function

([7], p. 2), H(R) on [A,co) is written as

(4.4) R_??_A

where limR,.c(R)=c(0<c<oo), B is a positive constant and ~(t) is a con

tinuous function such that limn.8(t)=0. In our situation we can choose

an appropriate H(R) satisfying (4.3) such that e(t)<1, c(R) is a positive

constant and A=B=1. Thus H'(R)=H(R)e(R)R1. Define e(u),& ~(u) by

f(u)e(u)V0, u)=--(~(u)A0). We have ~(u)_s~(u)--e_(u). Let et denote

the unit vector with i-th component 1. With the ei, we define a measure a

on Sd-1 in the following way: Support of a={±e:i=1,2,...,d} and

a({e~})==a({e~})=Jr1Jc'-1. On the interval (1,oo), define measures pe$ and

p_e{, i=1,2,•c,d, as follows:

(4.5) if

if

Then

(4.0)


We define a finite measure vp on Rd by

(4.7)

for

where 1E is the indicator function of E. Define a probability measure v1=co1vo, where c0 is the total measure of vo.

Then, we verify that v1 satisfies an analogue of (4.2). If R>1 , then

Since

we have

Hence v1 satisfies (4.2) with c•L' replaced by c-10c•L.

Next we prove that v1 satisfies (3.4). we have

the first term of which is equal to co1r~H(1). Noticing d-~e,A(de)=O and

usign (4.6) we see that the second term equals

Therefore xw1(dx)=cQ1r,H(R). Since Rv1(txJ>R)=c~'c'H(RX1+

o(1)), the equation (3.4) is proved.

360 TAKAAKI SHIMURA

Now we conclude that vl belongs to the strict domain of attraction of

p with normalizing constant c-10nh(n). Define v by v(E)=v,(co1E). Then v is a distribution satisfying the desired conditions.

In the case a~1, for any given slowly varying function h(x), we can

construct a distribution v with an appropriate tail behavior in a similar way to the case of a=1. In the case 0<a<1, this v is a distribution in SD(p) having the desired property by Proposition 4.1. In the case 1<a<2, the shifted measure v*&,,m (m is the mean of v) is the desired distribution by Proposition 4.2. Thus the proof is complete.

REMARK. In the case of index 2, the above theorem does not hold. In fact, let p be a Gaussian distribution (possibly degenerate) with mean 0 and

pro. Then for some v e SD(p) a slowly varying function h(x) is the slowly varying function part of normalizing constant for v if and only if

(4.8)

This condition means that h(x) is equivalent with an increasing slowly varying function. (This fact is proved for d=1 in [10] and for general d in

[9]).Next we study the problem whether shifts of a distribution in a strict

domain of attraction belong to a strict domain of attraction. In this consideration the slowly varying function part of normalizing constant plays an important role. For a non-trivial strictly stable law p with index 1 on Rd, we classify the strict domain of attraction of a strictly stable law with index 1 on Rd as follows.

or the limit does not exist

For ve3(Rd) and seRd, let vs denote v*o.

THEOREM 4.4. Each SD(p) (i=1,2,3) is non-empty. If veSD1(p), then,

for all seRd, vsESD1(p). If vESD2(p), then, for each seRd, vsESD(Pm(s)) for m(s)=c-1s where c=limR~.Rv(fxl>R). If vESD3(p), then, for every seRd\{0}

and every strictly stable law p,vsSD(p).

PROOF. The classification is made, based on the slowly varying function H(R) appearing in (4.2). Namely, Rv((xl>R) is a slowly varying function equivalent to H(R). Hence it follows from (4.3) that veSD1(p)(SD2(p),


SD3(p)) if and only if the slowly varying function h(R) associated with v tends to oo(toc(0<c<oo ), to 0 or the limit does not exist). Therefore all the cases are possible by Theorem 4.3. Let v be in the strict domain of attraction of the strictly stable law with characteristic function (3.1). We have

(4.9)

where

We can prove limR~.f2(R)=0 i=1,2,3 by calculation based on the fact that Rv(xf>R) is slowly varying. Since vs automatically satisfies (3.2) and (3.3), the rest of the proof follows from the equality (4.9).

REMARK. We can call SD2(p) the strict domain of normal attraction. In this case, let pi and ~2 be type-equivalent strictly stable laws with index 1. Then, for any veSD2(p1), there exists seRd uniquely such that v~ESD2(p2).

REMARK. Let p be a stable law with index a*1. In case 0<a<1, if veSD(p), then vsESD(p) for each seRd. In case 1<a<2, if veSD(p) and if p is a strictly stable law, then vsSD(u) for every sRd\{0}.

•˜ 5. Examples

There are some more interesting properties that are worth mentioning

regarding the strict domain of attraction with index 1. Two examples in

this section will serve to explain this situation. We fix a strictly stable law

p with index 1 on R1 and consider the difference between D(p) and SD(u).

Example 1 implies that the class of all shifts of distributions in SD(p) is a

proper subclass of D(p). Example 2 shows that a distribution in D(p) with

mean 0 does not necessarily belong to SD(p). These properties are peculiar

to the case of index 1.

EXAMPLE 1. Let v be a distribution with the following density function:

(5.1) p(x)=

c0x-2(1+(logx)-1) x_??_e0 |x|<ec0x-2 x_??_-e

362 TAKAAKI SHIMURA

where c0 is a normalizing constant. It is easy to prove that v belongs to the

domain of attraction of Cauchy distribution. For each seR1, we can prove

(5.2)

as

This shows that, for each s•‚0, vs does not belong to the strict domain of

attraction of Cauchy distribution.

EXAMPLE 2. Let vo be a distribution with the density function

(5.3) p0(x) =

c0(xlogx)-2(1+f(x)) x_??_e0 |x|<ec0(|x|log|x|)-2 x_??_-e

where f(x) is a continuous function on [e,oo) such that f(x)_??_-1 and lima~.f(x)=0 and co is a normalizing constant. It is easy to see that vg

belongs to the domain of attraction of Cauchy distribution and has finite mean m.

Let v be vo* Obviously, v belongs to the domain of attraction and

has mean 0. And we can prove the following equality if the limit in the right hand side exist:

(5.4)

Therefore, if we set f(x)=-2irr(clogx)_1(x~exp(2irTfcV0)), then (3.9) is satisfied. But, if f(x)=(logx)-1/2, then (3.9) is not satisfied. This means that the condition that a distribution belongs to the domain of attraction and has mean 0 does not guarantee that it belongs to the strict domain of attraction.

Acknowledgement The author wishes to express his gratitude to Professor Ken-iti Sato for his guidance and encouragement in the course of the research.

References

[1] N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular Variation, Encyclopedia of Math, and Its Appl., 1987.

[2] W. Feller, An Introduction to Probability Theory and Its Applications, 2nd ed., Vol. II, Wiley, New York, 1971.

[3] B. V. Gnedenko and A. N. Kolmogorov, Limit Distributions for Sums of Inde pendent Random Variables. (English translation) 2nd ed. Addison-Wesley, Cambridge, Mass, 1968.


[4] I. A. Ibragimov and Yu. V. Linnik, Independent and Stationary Sequences of Random Variables, Wolters-Noordhaff Publishing, Groningen, The Netherlands, 1971.

[5] P. Levy, Theorie de l'Addition des Variables Aleatoires. 2eme ed. Gauthier- Villars, Paris, 1954 (1ere ed., 1937).

[6] E. L. Rvaceva, On domain of attraction of multi-dimensional distributions, (English translation) Select. Transl. Math. Statist. and Probability, 2 (1954), 183

- 205. (L'vov. Gos. Univ. Uc. Zap. 29 Ser, Meh.-Mat. No. 6, 5-44)

[7] E. Seneta, Regularly Varying Functions, Lecture Notes in Mathematics 508, Springer-Verlag, Berlin, 1976.

[8] K. Sato, Infinitely Divisible Distributions, Seminar on Probability, 52, Kakuritsuron Seminar (in Japanese), 1981.

[9] T. Shimura, Decomposition of non-decreasing slowly varying functions and the domain of attraction of Gaussian distributions, submitted to J. Math. Soc. Japan.

[10] H. G. Tucker, Convolutions of distributions attracted to stable laws, Ann. Math. Statist., 39 (5) (1968), 1381-1390.

DEPARTMENT OF MATHEMATICS

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the strict domain of attraction of strictly stable law

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