for Â 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 †Â C
Eei.Â;X/ D exp
ei.Â;x/ 1 i Â; ŒŒx
.dx/ C i.Â; b/
2
Rd
(3.4)
for Â 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 Â 2 R.T/ ,
(
E exp i
X
)
Â.t/X.t/
(3.9)
t2T
D exp
1 T
Â †Â C
2
Z
ei.Â;x/
RT
1
i Â; ŒŒx
Á
.dx/ C i.Â; b/
with the notation
Â 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
Â †t1 ;:::;td Â
2
)
t1 ;:::;td .dx/
C i.Â; bt1 ;:::;td /
for Â 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
oÁ
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 †Â C i.Â; b/
Â.t/X.t/ D exp
(3.10)
2
t2T
for Â 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 Â 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
ŒŒx2 .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
ŒŒx2 .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 Â 2 R,
1 2
Â
2
EeiÂn D exp
2
n
;
C In .Â / 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 .Â / D
Z
iÂx
e
[1
jDnC1 Bj
Rnf0g
1 C iÂŒŒx
.ds; dx/ :
Since
ˇ iÂx
ˇe
1
ˇ
iÂ ŒŒxˇ Ä K.Â /ŒŒx2
for some K.Â / 2 .0; 1/, we conclude that
ˇ
ˇ
ˇIn .Â /ˇ Ä K.Â /
Z
Z
[1
jDnC1 Bj
Rnf0g
ŒŒx2 .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