Applied Mathematics and Computation 150 (2004) 337–345

www.elsevier.com/locate/amc

Inverse Laplace transformfor heavy-tailed distributions

Aldo Tagliani a,*, Yurayh Vel�asquez b

a Faculty of Economics, Trento University, 38100 Trento, Italyb Escuela de Ingeneria de Sistemas, Universidad Metropolitana, 52120 Caracas, Venezuela

Abstract

Laplace transform inversion on the real line of heavy-tailed (probability) density

functions is considered. The method assumes as known a finite set of fractional mo-

ments drawn from real values of the Laplace transform by fractional calculus. The

approximant is obtained by maximum entropy technique and leads to a finite gener-

alized Hausdorff moment problem. Directed divergence and L1-norm convergence are

proved.

� 2003 Elsevier Inc. All rights reserved.

Keywords: Fractional calculus; Fractional moments; Generalized Hausdorff moment problem;

Hankel matrix; Laplace transform inversion; Maximum entropy

1. Introduction

In this paper the attention is drawn to Laplace transform inversion

* Co

E-m

0096-3

doi:10.

F ðsÞ ¼Z 1

0

e�stf ðtÞdt ð1:1Þ

of heavy-tailed (probability) density functions f ðtÞ, which do not admit finitemean value EðX Þ ¼ �F 0ð0Þ ¼

R10

tf ðtÞdt, where Eð�Þ denotes the expectation.

Only real values F ðsÞ are used, as required in some physical problems. Then

several classical methods of inversion, requiring complex values F ðsÞ, are not

rresponding author.

ail addresses: ataglian@cs.unitn.it, taglian@cs.unitn.it (A. Tagliani).

003/$ - see front matter � 2003 Elsevier Inc. All rights reserved.

1016/S0096-3003(03)00235-2

338 A. Tagliani, Y. Vel�asquez / Appl. Math. Comput. 150 (2004) 337–345

applicable. The proposed inversion method leads to a generalized finite

Hausdorff moment problem EðX ajÞ ¼R10

taj f ðtÞdt ¼R 1

0ðx=ð1� xÞÞajwðxÞdx,

for some function wðxÞ in a new variable x 2 ½0; 1�. Fractional moments EðX ajÞ,06 aj < a� < 1 for some a� are drawn from real values of F ðsÞ through frac-

tional calculus [1]. If the given fractional moments EðX ajÞ are interpreted as a

priori information, we can refer to Maximum Entropy (ME) Principle [2] in the

choice of the approximant wMðxÞ of wðxÞ. It will be proved that wMðxÞ con-

verges to wðxÞ in entropy, in L1-norm and in directed divergence. As a conse-

quence, in the original domain t ¼ ½0;1Þ the approximating density fMðtÞconverges to f ðtÞ in L1-norm and in directed divergence too. Then, the ap-proximating density is particularly suitable for the computation of expected

values Ef ðgÞ, for some bounded function gðtÞ, as required in Applied Proba-

bility. The corresponding error resulting by replacing f ðtÞ with fMðtÞ is gov-

erned by the difference of entropies H ½wM � � H ½w�, where H ½w� is unknown,

while H ½wM � is calculated. Then the unknown difference H ½wM � � H ½w� may be

estimated through the calculated quantities H ½wj�, j ¼ 1; . . . ;M allowing the

calculation of Ef ðgÞ with a pre-fixed accuracy.

The present paper is a natural extension of the previous one [3] where thehypothesis EðX Þ ¼ �F 0ð0Þ ¼

R10

tf ðtÞdt finite was assumed.

2. Formulation of the problem

The main step consists in transforming the problem (1.1) into an equivalent

generalized Hausdorff moment problem as follows.

First of all, by fractional calculus, real values of F ðsÞ allow us to obtain

fractional moments [1]

EðX ajÞ ¼:

Z 1

0

taj f ðtÞdt ¼ ajCð1� ajÞ

Z 1

0

1� F ðsÞsajþ1

ds; 06 aj < a� < 1:

ð2:1Þ

Let be 06 a < a�, with EðX a� Þ < þ1 and fajg1j¼0 2 ½a; a�Þ, with a0 ¼ 0, a

sequence of infinite values. Then from (2.1) such a sequence guarantees the

existence of a unique density f ðtÞ, according to Lin result [4, Theorem 1]. By

the change of the variable t ¼ x=ð1� xÞ from (2.1) we have

EðX ajÞ ¼Z 1

0

taj f ðtÞdt ¼Z 1

0

x1� x

� �ajwðxÞdx ð2:2Þ

with wðxÞ ¼ ð1=ð1� xÞ2Þf ðx=ð1� xÞÞ, wð1Þ ¼ 0, x 2 ½0; 1Þ, which is equivalent

to a generalized finite Hausdorff moment problem. If fEðX ajÞgMj¼0 is a finite

sequence of fractional moments for some M P 1 and then such a sequence is

A. Tagliani, Y. Vel�asquez / Appl. Math. Comput. 150 (2004) 337–345 339

considered as given a priori information about wðxÞ, then we may refer to ME

technique. By maximizing Shannon-entropy

H ½w� ¼ �Z 1

0

wðxÞ lnwðxÞdx ð2:3Þ

constrained by (2.2), where j ¼ 0; . . . ;M the following approximant wMðxÞ ofwðxÞ is obtained [2]

wMðxÞ ¼ exp

�XMj¼0

kjx

1� x

� �aj!ð2:4Þ

with wMð1Þ ¼ 0 and kM P 0. Here ðk0; . . . ; kMÞ are Lagrange multipliers satis-

fying the constraints

EðX ajÞ ¼Z 1

0

x1� x

� �ajwMðxÞdx; j ¼ 0; . . . ;M : ð2:5Þ

wMðxÞ has Shannon entropy H ½wM � ¼ �R 1

0wMðxÞ lnwMðxÞdx ¼

PMj¼0 kjEðX ajÞ.

In the original domain t ¼ ½0;þ1Þ the approximant fMðtÞ of f ðtÞ assumes the

following analytical form

fMðtÞ ¼1

ð1þ tÞ2exp

�XMj¼0

kjtaj!: ð2:6Þ

In Appendix A we prove the following theorem

Theorem 2.1. If fajgMj¼0 are equispaced within an arbitrary interval ½0; a� < 1�,i.e. aj ¼ jða�=MÞ, with EðX a� Þ < 1, then ME approximant wMðxÞ converges inentropy to wðxÞ, i.e.,

limM!1

H ½wM � ¼ H ½w�: ð2:7Þ

If wðxÞ and wMðxÞ have the same fractional moments fEðX ajÞgMj¼0 then the fol-lowing result, relating directed divergence and entropy difference, follows

Iðw;wMÞ ¼:

Z 1

0

wðxÞ ln wðxÞwMðxÞ

dx ¼ H ½wM � � H ½w�; ð2:8Þ

where Iðw;wMÞ denotes the directed divergence. In fact

Iðw;wMÞ ¼Z 1

0

wðxÞ ln wðxÞwMðxÞ

dx ¼ �H ½w� þXMj¼0

kj

Z 1

0

ðx=ð1� xÞÞajwðxÞdx

¼ �H ½w� þXMj¼0

kjEðX ajÞ ¼ H ½wM � � H ½w�:

340 A. Tagliani, Y. Vel�asquez / Appl. Math. Comput. 150 (2004) 337–345

Another measure of distance between wðxÞ and wMðxÞ is given by variation

measure V ðw;wMÞ ¼R 1

0jwMðxÞ � wðxÞjdx. Directed divergence and variation

are related each other by the inequality [5]

Iðw;wMÞP1

2V 2ðw;wMÞ: ð2:9Þ

Entropy convergence (2.7) entails convergence in directed divergence from

(2.8) and convergence in L1-norm from (2.9).In Applied Probability one usually calculates expected values. If gðtÞ denotes

a bounded function, such that jgðtÞj6K taking into account (2.8), (2.9) and the

change of variable t ¼ x=ð1� xÞ one has

jEf ðgÞ � EfM ðgÞj ¼Z 1

0

gðtÞ f ðtÞð���� � fMðtÞÞdt

����6KZ 1

0

jf ðtÞ � fMðtÞjdt

¼ KZ 1

0

jwðxÞ � wMðxÞjdx6Kffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ðH ½wM � � H ½w�Þ

p:

ð2:10Þ

According to (2.7) and Theorem 2.1 about convergence in entropy of fMðxÞto f ðxÞ we are able to formulate the choice criterion of a1; . . . ; aM . The optimal

aj exponents are obtained as

fajgMj¼1 : H ½fM � ¼ minimum: ð2:11Þ

The sequence a1; . . . ; aM is optimal in the sense that it accelerates the conver-

gence of H ½fM � to H ½f �. Equivalently, it uses a minimum number of fractional

moments to reach a pre-fixed (even if unknown) gap H ½fM � � H ½f �.As a consequence, if the choice of equispaced fajgMj¼0, as required in The-

orem 2.1, guarantees entropy-convergence, then the choice (2.11) guarantees

entropy-convergence too.

From a computational point of view, the kj calculation leads to minimize the

following potential function Cðk1; . . . ; kMÞ [2], with

mink1;...;kM

Cðk1; . . . ; kMÞ ¼ mink1;...;kM

ln

Z 1

0

x1� x

� �aj "

� exp �XMj¼1

kjx

1� x

� �aj!dx

! þXMj¼1

kjEðX ajÞ#:

ð2:12Þ

From (2.11) and (2.12) the approximating density wMðxÞ is obtained through

two nested minimization procedures

A. Tagliani, Y. Vel�asquez / Appl. Math. Comput. 150 (2004) 337–345 341

wMðxÞ : mina1;...;aM

mink1;...;kM

ln

Z 1

0

x1� x

� �aj "

� exp

�XMj¼1

kjx

1� x

� �aj!dx

!þXMj¼1

kjEðX ajÞ#

ð2:13Þ

while k0 is calculated by imposing that the density wMðxÞ integrates to 1, i.e.

k0 ¼ ln

Z 1

0

exp

�XMj¼1

kjx

1� x

� �aj!dx

!:

Summarizing, the proposed inversion technique consists in two distinct steps:

(1) from real values of F ðsÞ, obtained analytically or numerically, one calcu-

lates fractional moments EðX ajÞ by (2.1), as required in (2.13);

(2) ME approximant wMðxÞ is obtained by (2.13) from which the approximant

fMðtÞ of f ðtÞ as in (2.6).

3. Convergence in the original domain t ¼ [0;‘)

In Section 2 we proved that the choice fajgMj¼0, according to (2.11), guar-

antees that wMðxÞ converges to wðxÞ in directed divergence, in L1-norm and in

entropy. In the original domain t ¼ x=ð1� xÞ the above kinds of convergence

entail

(a) From Iðw;wMÞ ¼R 1

0wðxÞ ln wðxÞ

wM ðxÞ dx ¼ H ½wM � � H ½w� ! 0 as M ! 1, we

have

Z 1

0

wðxÞ ln wðxÞwMðxÞ

dx ¼Z 1

0

f ðtÞ ln f ðtÞfMðtÞ

dt ¼ Iðf ; fMÞ ! 0 ð3:1Þ

i.e., fMðtÞ converges to f ðtÞ in directed divergence.

(b) FromR 1

0jwMðxÞ � wðxÞjdx ! 0 as M ! 1, we have

Z 1

0

jwMðxÞ � wðxÞjdx ¼Z 1

0

jfMðtÞ � f ðtÞjdt ! 0 ð3:2Þ

i.e., fMðtÞ converges to f ðtÞ in L1-norm.

(c) Entropy convergence H ½fM � ! H ½f � is proved under mild restrictions.

Proposition. If EðX 2aÞ ¼R10

t2af ðtÞdt, with ’ 0:76 < 2a < 1, exists thenH ½fM � ! H ½f �, as M ! 1.

342 A. Tagliani, Y. Vel�asquez / Appl. Math. Comput. 150 (2004) 337–345

Proof. From

Table

Entrop

of com

M

1

2

3

4

H ½w� ¼ H ½f � � 2

Z 1

0

lnð1þ tÞf ðtÞdt

and analogously for H ½wM � and H ½fM � we have

H ½fM � � H ½f � ¼ H ½wM � � H ½w� þ 2

Z 1

0

lnð1þ tÞ½fMðtÞ � f ðtÞ�dt

and then

jH ½fM � � H ½f �j6 jH ½wM � � H ½w�j þ 2

Z 1

0

lnð1þ tÞjfMðtÞ � f ðtÞjdt: ð3:3Þ

If aP 0:38 then lnð1þ tÞ < ta 8t > 0. Then from Schwarz inequality we have

Z 1

0

lnð1þ tÞjfMðtÞ � f ðtÞjdt6Z 1

0

tajfMðtÞ � f ðtÞjdt

6

Z 1

0

t2ajfMðtÞ�

� f ðtÞjdt�1=2 Z 1

0

jfMðtÞ�

� f ðtÞjdt�1=2

ð3:4Þ

R10

t2ajfMðtÞ � f ðtÞjdt is bounded 8M : it is enough to consider equispaced

fajgMj¼0, with aM > 2a. Then by taking into account (2.7), from (3.3) and (3.4)

we have

limM!1

H ½fM � ¼ H ½f �: � ð3:5Þ

4. Numerical results

Let us consider the Laplace transform

F ðsÞ ¼ 2

pcosðsÞ p

2

hh� SiðsÞ

i� sinðsÞCiðsÞ

i

where SiðsÞ and CiðsÞ denote sine and cosine integral respectively. Then

f ðtÞ ¼ ð2=pÞð1=ð1þ t2ÞÞ (Cauchy distribution) with H ½f � ’ 1:83787707,H ½w� ’ �0:02151373, while H ½wM � is calculated by (2.13). EðX aÞ are calculatedby (2.1) or, for simplicity, EðX aÞ ¼ 1= cosðp

2aÞ, 06 a < 1. In Table 1 are

reported

1

y difference, directed divergence and L1-norm of distributions having an increasing number

mon fractional moments

H ½fM � � H ½f � Iðf ; fM Þ kfM � f k10.4180E)1 0.1931E)1 0.1682E)00.2175E)1 0.8857E)2 0.1102E)0

)0.3047E)1 0.5828E)2 0.8795E)10.3510E)1 0.3958E)2 0.6635E)1

A. Tagliani, Y. Vel�asquez / Appl. Math. Comput. 150 (2004) 337–345 343

1. H ½fM � � H ½f �;2. Iðf ; fMÞ ¼

R10

f ðtÞ ln f ðtÞfM ðtÞ dt ¼

R 1

0wðxÞ ln wðxÞ

wM ðxÞ dx ¼ H ½wM � � H ½w�;3.R10

jf ðtÞ � fMðtÞjdt ¼R 1

0jwðxÞ � wMðxÞjdx

for an increasing number of expected values. Inspection of Table 1 reveals:

(1) Convergence in directed divergence, according to convergence in entropy

of wMðxÞ to wðxÞ.(2) A slow and oscillating convergence in entropy of fMðtÞ to f ðtÞ. When

M ¼ 3, H ½fM � � H ½f � < 0 simply means that fMðtÞ, as in (2.6), is not MEdistribution under the constraints fEðX ajÞgMj¼0.

Appendix A. Entropy convergence

A.1. Some background

Let

lj ¼: EðX ajÞ ¼Z 1

0

taj f ðtÞdt; j ¼ 0; . . . ;M ðA:1Þ

EðX a� Þ < 1, 0 < a� < 1 and aj ¼ jða�=MÞ, j ¼ 0; . . . ;M a sequence of equi-

spaced points 2 ½0; a��. When M ! 1, fEðX ajÞgMj¼0 characterize a unique

density f ðtÞ [4, Theorem 1]. With the change of variable x ¼ ta�=M (A.1) be-

comes

(1) a finite Stieltjes moment problem;

(2) when M ! 1 a determinate Stieltjes moment problem.

Introducing the following symmetric definite positive Hankel matrices

D0 ¼ l0; D2 ¼l0 l1

l1 l2

� �; . . . ;D2M ¼

l0 � � � lM

..

.� � � ..

.

lM � � � l2M

264

375 ðA:2Þ

then Stieltjes moment problem determinacy entails that the maximum mass

qðtÞ which can be concentrated at any real point t is equal to zero [6, Corollary

2.8]. In particular, at t ¼ 0 we have

0 ¼ qð0Þ ¼ limi!1

qð0Þi ¼:

jD2ijl2 � � � liþ1

..

.� � � ..

.

liþ1 � � � l2i

��������������¼ lim

i!1l0 � l�ðiÞ

0

� �; ðA:3Þ

344 A. Tagliani, Y. Vel�asquez / Appl. Math. Comput. 150 (2004) 337–345

where qð0Þi indicates the largest mass which can be concentrated at a given point

t ¼ 0 by any solution of a reduced moment problem of order P i and l�ðiÞ0

indicates the minimum value of l0 once assigned the first 2i moments.

Let us fix fl0; . . . ; li�1; liþ1; . . . ; lMg while only li, i ¼ 0; . . . ;M varies con-

tinuously. From

lj ¼: EðX ajÞ ¼Z 1

0

x1� x

� �ajwMðxÞdx; j ¼ 0; . . . ;M ðA:4Þ

with wMðxÞ ¼ expð�PM

j¼0 kjð x1�xÞ

ajÞ and aj ¼ jða�=MÞ we have

D2M

dk0=dli

..

.

dkM=dli

264

375 ¼ �eiþ1; ðA:5Þ

where eiþ1 is the canonical unit vector 2 RMþ1, from which

0 <dk0dli

; . . . ;dkMdli

� �D2M

dk0=dli

..

.

dkM=dli

264

375 ¼ � dk0

dli; . . . ;

dkMdli

� �eiþ1 ¼ � dki

dli8i

ðA:6Þ

A.2. Entropy convergence

The following theorem holds.

Theorem A.1. If aj ¼ jða�=MÞ, j ¼ 0; . . . ;M then

limM!1

H ½wM � ¼: �Z 1

0

wMðxÞ lnwMðxÞdx ¼ H ½w�

¼: �Z 1

0

wðxÞ lnwðxÞdx: ðA:7Þ

Proof. From (A.7) we have

H ½wM � ¼XMj¼0

kjlj: ðA:8Þ

Let us consider (A.8). When only l0 varies continuously, taking into account

(A.3)–(A.8) we have

d

dl0

H ½wM � ¼XMj¼0

ljdkjdl0

þ k0 ¼ k0 � 1;

A. Tagliani, Y. Vel�asquez / Appl. Math. Comput. 150 (2004) 337–345 345

d2

dl20

H ½wM � ¼dk0dl0

¼ �

l2 � � � lMþ1

..

. � � � ...

lMþ1 � � � l2M

��������������

jD2M j¼ � 1

l0 � l�ðMÞ0

< 0:

Thus H ½wM � is a concave differentiable function of l0. When l0 ! l�ðMÞ0 then

H ½wM � ! �1, whilst at l0 it holds H ½wM � > H ½w�, being wMðxÞ the ME density

once assigned (l0; . . . ; lM ). Besides, when M ! 1 then l�ðMÞ0 ! l0. So the

theorem is proved. �

Entropy convergence is guaranteed and accelerated whenever equispaced

nodes aj are replaced by optimal nodes (2.11).

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