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1 Infinitely Divisible Random Variables, Vectors, and Processes

1 Infinitely Divisible Random Variables, Vectors, and Processes

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74



3 Infinitely Divisible Processes



for  2 R, where for x 2 R, its truncation ŒŒx is defined by

ŒŒx D



8




if jxj Ä 1;

1 if x < 1;

:

1 if x > 1:



(3.2)



The triple . 2 ; ; b/ is called the characteristic triple of the infinitely divisible

random variable X, and it consists of 2

0, a measure

on R satisfying

f0g D 0 and

Z

R



1 ^ x2



.dx/ < 1 ;



and finally, b 2 R. See Sato (1999).

The basic examples of infinitely divisible random variables are the normal

random variable and the compound Poisson random variable of the following

example.

Example 3.1.1. If N is a mean Poisson random variable independent of an i.i.d.

sequence Y1 ; Y2 ; : : : with a common distribution FY , then

XD



N

X



Yi



(3.3)



iD1



is an infinitely divisible random vector because its characteristic function is of the

form (3.1) with 2 D 0,

D FY , and b D EŒŒY1 . An infinitely divisible

random variable with a representation of the form (3.1.1) is said to be compound

Poisson. Note that a compound Poisson random variable has a finite measure in

its characteristic triple.

Other examples of one-dimensional infinitely divisible random variables include

the gamma random variable and the geometric (and, more generally, the negative

binomial) random variable, as can be verified by identifying, in each of these examples, the characteristic triple. See Exercise 3.8.1.

The entry 2 in the characteristic triple of an infinitely divisible random variable

X is the variance of the Gaussian component of X; if 2 D 0, we say that

the infinitely divisible random variable has no Gaussian component. The entry

in the characteristic triple of an infinitely divisible random variable X is the

Lévy measure of X; it describes the Poisson component of the infinitely divisible

random variable; if

D 0, the infinitely divisible random variable has no

Poisson component and hence is a Gaussian random variable. The entry b in

the characteristic triple of an infinitely divisible random variable X is sometimes

referred to as the shift component of X; this, however, has to be taken with a grain

of salt, since b interacts with the truncation ŒŒ  in the representation (3.1) of the



3.1 Infinitely Divisible Random Variables, Vectors, and Processes



75



characteristic function. In certain special cases, the characteristic function of an

infinitely divisible random variable has slightly different representations, in which

the role of a “shift” parameter is clearer.

Example 3.1.2. A Lévy motion

A Lévy motion (also known as a Lévy process) is continuous in a probability

stochastic process with stationary and independent increments, X.t/; t 2 R , such

that X.0/ D 0 a.s. Note that it is also common to define a Lévy process only

on the positive half-line, as X.t/; t

0 , satisfying the same requirements. The

definition on the entire real line is more natural and convenient for our purposes.

The finite-dimensional distributions of a Lévy process are completely determined by

its one-dimensional marginal distribution at time 1, which is necessarily infinitely

divisible (in one dimension). In fact, there is a one-to-one correspondence between

the laws of Lévy processes and the laws of one-dimensional infinitely divisible

random variables that are the values of the Lévy processes at time 1. See Sato (1999).

It is easy to see that a Lévy process is an infinitely divisible stochastic process.

Indeed, let n D 1; 2; : : :. Since X D X.1/ is a one-dimensional infinitely divisible

d

random variable, there is an infinitely divisible random variable Y such that X D

Y1 C : : : C Yn , where Y1 ; : : : ; Yn are i.i.d. copies of Y. Let Y.t/; t 2 R be a Lévy

d



process such that Y.1/ D Y, and let Yj .t/; t 2 R ; j D 1; : : : ; n, be i.i.d. copies

Pd

of Y.t/; t 2 R . Then the stochastic process

jD1 Yj .t/; t 2 R is continuous

in probability, has stationary and independent increments, and vanishes at time 0.

Therefore, it is a Lévy process. Since

d

X



d



Yj .1/ D



jD1



d

X



d



d



Yj D X D X.1/



jD1



by construction, we conclude that

1

0

d

X

d

@

Yj .t/; t 2 RA D X.t/; t 2 R :

jD1



Therefore, X.t/; t 2 R is an infinitely divisible stochastic process.

A Brownian motion is a particular Lévy process for which X.1/ has a normal

distribution.

An infinitely divisible stochastic process corresponding to a finite parameter

set T is a (finite-dimensional) infinitely divisible random vector. The distribution

of an infinitely divisible random vector, say X D .X .1/ ; : : : ; X .d/ /, is once again

uniquely determined by a characteristic triple †; ; b , where this time, † is a

d d nonnegative definite matrix, and a measure on Rd such that f0g D 0 and

Z

Rd



1 ^ kxk2



.dx/ < 1 :



76



3 Infinitely Divisible Processes



Finally, b 2 Rd . The characteristic function of an infinitely divisible random vector,

say X, with a characteristic triple †; ; b is given by

Z

Á

1 T

 †Â C

Eei.Â;X/ D exp

ei.Â;x/ 1 i Â; ŒŒx

.dx/ C i.Â; b/

2

Rd

(3.4)

for  2 Rd , where the truncation of a vector is defined componentwise: for x D

x.1/ ; : : : ; x.d/ ,

ŒŒx D ŒŒx.1/ ; : : : ; ŒŒx.d/  :

The role of the entries in the characteristic triple of an infinitely divisible random

vector X is similar to their role in the one-dimensional case. The entry † is the

covariance matrix of the Gaussian component of X, the measure describes the

Poisson component of X, and the vector b can be (imprecisely) thought of as

the shift vector of X.

The following characterization of an infinitely divisible stochastic process is

almost immediate; see Exercise 3.8.5.

Proposition 3.1.3. A stochastic process X.t/; t 2 T is infinitely divisible if and

only if all of its finite-dimensional distributions are infinitely divisible.

The most useful description of infinitely divisible processes is through their own

characteristic triples. Such triples are similar in nature to the characteristic triples

of one-dimensional infinitely divisible random variables and of infinitely divisible

random vectors, but this time, they “live on the appropriate function spaces.” In the

description below, we follow Rosi´nski (2007).

For a parameter space T, let RT be the collection of all real-valued functions

on T. We transform RT into a measurable space by endowing it with the cylindrical

-field. A measure on RT is said to be a Lévy measure if the two conditions stated

below hold. The first condition is as follows:

Condition 3.1.4.

Z

RT



ŒŒx.t/2 .dx/ < 1 for every t 2 T.



(3.5)



As in the finite-dimensional case, the truncation of a function in (3.5) is understood

componentwise: for x D x.t/; t 2 T 2 RT , ŒŒx 2 RT is defined by

ŒŒx.t/ D ŒŒx.t/; t 2 T :

The second condition will be stated separately in two cases: when the parameter

space T is countable (or finite), and when it is uncountable.

Condition 3.1.5 (Countable space T).

Á

x 2 RT W x.t/ D 0 for all t 2 T D 0 :



(3.6)



3.1 Infinitely Divisible Random Variables, Vectors, and Processes



77



Condition 3.1.6 (Uncountable space T). For every countable subset T1 of T, such

that

Á

(3.7)

x 2 RT W x.t/ D 0 for all t 2 T1 > 0;

there is t0 2 T1c such that

Á

x 2 RT W x.t/ D 0 for all t 2 T1 ; x.t0 / 6D 0 > 0 :



(3.8)



To state the theorem characterizing infinitely divisible stochastic processes, we

introduce the following notation: let

n

o

R.T/ D x 2 RT W x.t/ D 0 for all but finitely many t 2 T

be the collection of functions on T vanishing outside

P of a finite set. Note the obvious

fact that if x 2 R.T/ and y 2 RT , then the sum t2T x.t/y.t/ makes perfect sense

because the (potentially uncountable) number of terms in the sum is really only

finite, corresponding P

to theP

set of nonzero coordinates of x. Similarly, if a 2 RT T ,

then the double sum s2T t2T a.s; t/x.s/x.t/ makes perfect sense as well.

Theorem 3.1.7. A stochastic process X.t/; t 2 T is infinitely divisible if and only

if there exists a uniquely determined triple †; ; b such that for every  2 R.T/ ,

(

E exp i



X



)

Â.t/X.t/



(3.9)



t2T



D exp



1 T

 †Â C

2



Z

ei.Â;x/

RT



1



i Â; ŒŒx



Á



.dx/ C i.Â; b/



with the notation

 T †Â D



XX

s2T t2T



†.s; t/Â.s/Â.t/; .Â; y/ D



X



Â.t/y.t/; y 2 RT :



t2T



In (3.9):

• † D †.s; t/; s; t 2 T is a nonnegative definite function on T;



is a Lévy measure on T;

• b 2 RT .

See Rosi´nski (2007) for a proof. The sufficiency part of Theorem 3.1.7 is clear: if

a stochastic process X.t/; t 2 T satisfies (3.9) for some triple †; ; b , as in the

theorem, then for every finite subset ft1 ; : : : ; td g of T, the d-dimensional random

vector X. t1 /; : : : ; X.td / has the characteristic function



78



3 Infinitely Divisible Processes



E exp



8

< X



Âj X.tj /



:



jD1;:::;d



9

=

;



(

D exp



Z

C



Rft1 ;:::;td g



ei.Â;x/



1



i Â; ŒŒx



Á



1 T

 †t1 ;:::;td Â

2

)



t1 ;:::;td .dx/



C i.Â; bt1 ;:::;td /



for  D .Â1 ; : : : ; Âd / 2 Rd , with †t1 ;:::;td the restriction of † from T T to

ft1 ; : : : ; td g ft1 ; : : : ; td g, bt1 ;:::;td the restriction of b from T to ft1 ; : : : ; td g, and

ft1 ;:::;td g

defined by

t1 ;:::;td a measure on R

t1 ;:::;td .B/



D



Á

x 2 RT W x.t1 /; : : : ; x.td / 2 B n f0g



for Borel sets B

Rft1 ;:::;td g . Therefore, X.t1 /; : : : ; X.td / is a d-dimensional

infinitely divisible random vector, and so all finite-dimensional distributions of the

process X.t/; t 2 T are infinitely divisible. By Proposition 3.1.3, one concludes

that X.t/; t 2 T is an infinitely divisible process.

As in the finite-dimensional case, the triple †; ; b appearing in Theorem 3.1.7

is referred to as the characteristic triple of the infinitely divisible process, and the

representation (3.9) its Lévy–Khinchine representation.

Remark 3.1.8. Conditions 3.1.5 and 3.1.6 on a Lévy measure on T ensure that the

measure does not put “redundant” weight on functions with “many” zero coordinates. This guarantees (by Theorem 3.1.7) that an infinitely divisible stochastic

process has a unique Lévy measure.

Let, on the other hand, X.t/; t 2 T be an infinitely divisible stochastic process

and suppose that †; ; b is a triple as in Theorem 3.1.7, except that the measure

is assumed to satisfy only Condition 3.1.4 but not necessarily conditions 3.1.5

and 3.1.6. In such a case we will say that is a weak Lévy measure of the process

X.t/; t 2 T , and †; ; b is a weak characteristic triple.

“Redundant weight on zeros” of a weak Lévy measure does not play any role in

the representation (3.9) of the characteristic function of the process. For example,

suppose that T is uncountable and there exists a countable set T1 such that (3.7)

holds but (3.8) fails for every t0 2 T1c . We can then remove some “redundant zeros”

by defining a measure 1 on RT by

1 .A/



D





n

A \ x 2 RT W x.t/ 6D 0 for some t 2 T1 ; A a cylindrical set.



The triple †; 1 ; b clearly still satisfies (3.9). Therefore, a weak Lévy measure is

not unique.

In some cases, we will find it convenient to work with weak Lévy measures of

infinitely divisible stochastic processes. Apart from lack of uniqueness, their role in

describing the structure of infinitely divisible processes is identical to that of “true”

Lévy measures.



3.1 Infinitely Divisible Random Variables, Vectors, and Processes



79



Example 3.1.9. A Gaussian process is a stochastic process X.t/; t 2 T whose

finite-dimensional distributions are multivariate normal distributions. These are

infinitely divisible stochastic processes with characteristic triples in which the Lévy

measure vanishes. That is, the characteristic function of a Gaussian process can

be written in the form

(

)

X

1 T

E exp

 †Â C i.Â; b/

Â.t/X.t/ D exp

(3.10)

2

t2T

for  2 R.T/ . Then † is the covariance function of the process, and b is its mean

function.

The following proposition lists elementary properties of the characteristic triples

of infinitely divisible stochastic processes. We leave the proof for Exercise 3.8.10.

Proposition 3.1.10. (i) If X.t/; t 2 T is an infinitely divisible stochastic process

with a characteristic triple †; ; b and a 2 R n f0g, then aX.t/; t 2 T is an

infinitely divisible stochastic process with the characteristic triple †a ; a ; ba ,

where

Z

†a D a2 †; a . / D a 1 ; ba D ab C

aŒŒx ŒŒax .dx/ :

RT



(ii) If X1 .t/; t 2 T and X2 .t/; t 2 T are independent infinitely divisible

stochastic processes with respective characteristic triples †i ; i ; bi , i D 1; 2,

then X1 .t/ C X2 .t/; t 2 T is an infinitely divisible stochastic process with the

characteristic triple †1 C †2 ; 1 C 2 ; b1 C b2 .

Example 3.1.11. A stochastic process X.t/; t 2 T is called ˛-stable if it has the

following property: for every n 1, there is a (nonrandom) function .cn .t/; t 2 T/

such that

0

1

n

X

d

X.t/; t 2 T D @n 1=˛

(3.11)

Xj .t/ C cn .t/; t 2 T A ;

jD1



where Xj .t/; t 2 T ; j D 1; : : : ; n, are i.i.d. copies of X.t/; t 2 T . It is called

strictly ˛-stable if cn .t/ Á 0 for every n 1. It is called symmetric ˛-stable (often

abbreviated to S˛S) if it is ˛-stable and symmetric. It is clear that a symmetric

˛-stable process is also strictly ˛-stable.

The definition immediately implies that every ˛-stable process is infinitely

divisible. If †; ; b is the characteristic triple of the process X.t/; t 2 T ,

then, by Proposition 3.1.10, the stochastic process on the right-hand side of (3.11)

is infinitely divisible with the characteristic triple n1 2=˛ †; n .n1=˛ /; bn for



80



3 Infinitely Divisible Processes



some function bn 2 RT . It follows from Theorem 3.1.7 that a stochastic process

X.t/; t 2 T is ˛-stable if and only if it is infinitely divisible and its characteristic

triple †; ; b satisfies

† D n1



2=˛



† and



D n .n1=˛ /



(3.12)



for every n D 1; 2; : : :.

It follows immediately from (3.12) that † D 0 unless ˛ D 2. Further, suppose

that 6D 0. Then there exists t0 2 T such that

Á

x 2 RT W jx.t0 /j > a > 0



h.a/ D



for some a > 0. Clearly, h is a nonincreasing function. Since it follows from (3.12)

that h an1=˛ D n 1 h.a/, we conclude that necessarily ˛ > 0, and further, this

relation means that lima!1 a˛ h.a/ D 1. Since

Z

RT



ŒŒx.t0 /2 .dx/



Z

0



1



yh.y/ dy ;



Condition 3.1.4 on a Lévy measure implies that ˛ < 2.

That is, a nondeterministic ˛-stable process can exist only for 0 < ˛ Ä 2. If

˛D2, then the Lévy measure of the process vanishes, and the process is the Gaussian

process of Example 3.1.9.

If 0 < ˛ < 2, then the function † in the characteristic triple of the process must

vanish, while (3.12) implies that the Lévy measure of the process must have the

scaling property

.c / D c



˛



for each c > 0.



(3.13)



See Exercise 3.8.11.

Conversely, every infinitely divisible process with † D 0 and Lévy measure

that scales as in (3.13) is ˛-stable. One usually constructs ˛-stable processes as

stochastic integrals as in Section 3.3.

We summarize the discussion of this section by recording the following immediate but important corollary of Theorem 3.1.7.

Corollary 3.1.12. A stochastic process X.t/; t 2 T is infinitely divisible if and

only if it has a unique decomposition in law

d



X.t/; t 2 T D G.t/; t 2 T C Y.t/; t 2 T ;

where G.t/; t 2 T and Y.t/; t 2 T are independent stochastic processes, with

G.t/; t 2 T a centered Gaussian process and Y.t/; t 2 T has a characteristic

function of the form



3.2 Infinitely Divisible Random Measures



(

E exp



X



)



Z



Â.t/Y.t/ D exp



t2T



for  2 R.T/ , where



81



1



ei.Â;x/

RT



i Â; ŒŒx



Á



.dx/ C i.Â; b/



is a Lévy measure on T, and b is a real-valued function on T.



If X.t/; t 2 T has a characteristic function given by (3.9), then † is the

covariance function of the Gaussian process G.t/; t 2 T (the Gaussian component

of X.t/; t 2 T ), and the rest of the characteristic triple †; ; b determines the

law of the process Y.t/; t 2 T . The latter process is an infinitely divisible process

without a Gaussian component (the Poisson component of X.t/; t 2 T plus a

deterministic “shift”).



3.2 Infinitely Divisible Random Measures

An infinitely divisible random measure is the single most important infinitely

divisible stochastic process, and it often serves as the basic ingredient in constructing other infinitely divisible stochastic processes. In order to define an infinitely

divisible random measure, we start with a measurable space .S; S/. We need three

measures as ingredients:

• a -finite measure

• a measure on S



on S;

R n f0g such that the measure

Z Z

m0 .B/ WD

B



Rnf0g



ŒŒx2 .ds; dx/; B 2 S



(3.14)



is -finite;

• a -finite signed measure ˇ on S (see Section 10.3).

Let S0 denote the collection of sets B in S satisfying

m.B/ WD .B/ C kˇk.B/ C m0 .B/ < 1 :



(3.15)



For sets B1 ; B2 2 S0 , define

† B1 ; B2 D



B1 \ B2 :



Note that for every d D 1; 2; : : :, a1 ; : : : ; ad 2 R, and B1 ; : : : ; Bd 2 S0 ,

12

0

Z X

d X

d

d

X

aj ak † Bj ; Bk D @

aj 1 s 2 Bj A .ds/ 0 :

jD1 kD1



S



jD1



Therefore, † is a nonnegative definite function on S0 .



(3.16)



82



3 Infinitely Divisible Processes



Á

R n f0g ! RS0 be a map defined by



Next, let ˆ W S



ˆ.s; x/.B/ D x1.s 2 B/; B 2 S0

for s 2 S and x 2 R n f0g. Clearly, ˆ is a measurable map, and it defines a measure

on RS0 by

ıˆ



D

We claim that



1



:



(3.17)



is a Lévy measure on S0 . Note, first of all, that for every B 2 S0 ,

Z

RS0



ŒŒx.B/2 .dx/ D



Z Z

B



Rnf0g



ŒŒx2 .ds; dx/ < 1



by the definition (3.15) of the collection S0 . Therefore, the measure satisfies Condition 3.1.4. If the collection S0 is countable, we also need to check Condition 3.1.5.

Since the measure m in (3.15) is -finite, the countable collection S0 covers S. Then

Á

x 2 RS0 W x.B/ D 0 for all B 2 S0 D



Á

.s; x/ W s … [B2S0 B D



; D 0;



and so Condition 3.1.5 holds. If the collection S0 is uncountable, we need to check

Condition 3.1.6. Suppose that Bj 2 S0 ; j D 1; 2; : : :, are such that

Á

x 2 RS0 W x.Bj / D 0 for all j D 1; 2; : : : > 0 :

Then



Á

.s; x/ W s … [jD1;2;::: Bj > 0 ;



which is equivalent to saying that for the measure m0 in (3.14),

Á

c

m0 [jD1;2;::: Bj

> 0:

Since the measures m0 and m are -finite, there is a set B in S0 such that B

c

[jD1;2;::: Bj and m0 .B/ > 0. Then also

Á

.s; x/ W s … [jD1;2;::: Bj ; s 2 B > 0 ;

which means that



Á

x 2 RS0 W x.Bj / D 0 for all j D 1; 2; : : :; x.B/ > 0 > 0 :



Therefore, Condition 3.1.6 holds, and the measure

on S0 .



in (3.17) is a Lévy measure



3.2 Infinitely Divisible Random Measures



83



Finally, define b 2 RS0 by

b.B/ D ˇ.B/; B 2 S0 :



(3.18)



The infinitely divisible stochastic process M D M.B/; B 2 S0 with the

characteristic triple †; ; b defined by (3.16), (3.17) and (3.18) is called an

infinitely divisible random measure on .S; S/ with Gaussian variance measure ,

Lévy measure , and shift measure ˇ.

The basic properties of an infinitely divisible random measure are described in

the following proposition.

Proposition 3.2.1. (i) For every B 2 S0 , M.B/ is an infinitely divisible random

variable with a characteristic triplet . B2 ; B ; bB /, where

2

B



D .B/;



B.



/ D .B



/; ˇB D ˇ.B/ :



(ii) An infinitely divisible random measure is independently scattered. That is,

M.B1 /; : : : ; M.Bd / are independent for every collection of disjoint sets

B1 ; : : : ; Bd 2 S0 .



(3.19)



(iii) An infinitely divisible random measure is -additive. That is, for each choice

of disjoint sets .Bj / S0 such that [j Bj 2 S0 ,

Á X

M.Bj / a.s.

M [j Bj D



(3.20)



j



(note that the exceptional set in (3.20) depends, in general, on the choice of

.Bj / S0 ).

Proof. Part (i) follows from the construction of the stochastic process M D

M.B/; B 2 S0 and Theorem 3.1.7. For part (ii), we use the fact that the

components of an infinitely divisible random vector are independent if and only

if every pair of the components is independent (see Sato (1999)). Therefore, it is

enough to prove part (i) in the case d D 2, and then the statement follows by noticing

that for disjoint sets B1 ; B2 2 S0 , the Gaussian component of the infinitely divisible

random vector N.B1 /; N.B2 / has vanishing covariance, while the bivariate Lévy

measure is concentrated on the axes in the plane. This implies the independence of

N.B1 / and N.B2 /; see Sato (1999).

For part (iii) of the proposition, it is enough to prove that

n WD



n

X

jD1



M.Bj /



Á

!0

M [1

B

j

jD1



(3.21)



84



3 Infinitely Divisible Processes



in probability as n ! 1, which will, in turn, follow if we check that the

characteristic function of the difference n on the left-hand side converges to the

constant function. By (3.9), for  2 R,

1 2

Â

2



Eein D exp



2

n



;



C In . / C iÂbn



where by (3.16),

2

n



D



n

X



[1

jD1 Bj



.Bj / C



Á



2



jD1



n

Ái

h

X

D

Bj \ [1

B

k

kD1



Á

[1

!0

B

j

jDnC1



jD1



as n ! 1 because [j Bj 2 S0 . Further, by (3.18),

bn D



n

X



ˇ.Bj /



Á

ˇ [1

jD1 Bj D



Á

ˇ [1

jDnC1 Bj ! 0



jD1



as n ! 1, also because [j Bj 2 S0 . Finally, by (3.17),

Z

In . / D



Z

iÂx



e



[1

jDnC1 Bj



Rnf0g



1 C iÂŒŒx



.ds; dx/ :



Since

ˇ iÂx

ˇe



1



ˇ

i ŒŒxˇ Ä K. /ŒŒx2



for some K. / 2 .0; 1/, we conclude that

ˇ

ˇ

ˇIn . /ˇ Ä K. /



Z



Z

[1

jDnC1 Bj



Rnf0g



ŒŒx2 .ds; dx/ ! 0



as n ! 1, once again because [j Bj 2 S0 , whence the -additivity of the random

measure.

The Lévy measure of an infinitely divisible random measure can also be

represented in a disintegrated form, given in the following proposition. This

representation has the advantage of providing additional intuition into the structure

of infinitely divisible random measures.

Proposition 3.2.2. There exists a family of one-dimensional Lévy measures

.s; /; s 2 S that is measurable in the sense that for every Borel set A 2 R n f0g,

the function .s; A/; s 2 S is measurable, and such that for each such A and

B 2 S0 ,

Z

.s; A/ m.ds/ :

(3.22)

B A D

B



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