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Elect. Comm. in Probab. 6 (2001) 15–29

ELECTRONIC
COMMUNICATIONS
in PROBABILITY

L1–NORM OF INFINITELY DIVISIBLE
RANDOM VECTORS AND CERTAIN
STOCHASTIC INTEGRALS
MICHAEL B. MARCUS
Department of Mathematics
The City College of CUNY, New York, NY 10031
email: mbmarcus@earthlink.net
´
JAN ROSINSKI
Department of Mathematics
University of Tennessee, Knoxville, TN 39996
email: rosinski@math.utk.edu
submitted September 10, 2000 Final version accepted January 10, 2001
AMS 2000 Subject classification: 60E07, 60E15, 60H05
Infinitely divisible random variables, stochastic integrals

Abstract
Equivalent upper and lower bounds for the L1 norm of Hilbert space valued infinitely
divisible random variables are obtained and used to find bounds for different types of
stochastic integrals.

1

L1–norm of infinitely divisible random vectors

Let X be an infinitely divisible random vector in a separable Hilbert space H. (See
e.g. [3], [7].) Assume that EkXk < ∞, EX = 0 and that X does not have a Gaussian
component. The characteristic function of X can be written in the form

Z
ihy,xi
(e
− 1 − ihy, xi) Q(dx)
(1)
E exp ihy, Xi = exp
H

where Q is a unique σ–finite Borel measure on H with Q({0}) = 0 and
Z
min{kxk2 , kxk} Q(dx) < ∞.

(2)

H

Q is called the L´evy measure of X. It is one of the principle entities used to describe
the distribution of X.
We obtain bounds for EkXk in terms of a functional of Q. Assume that Q(H) > 0
and let l = l(Q) > 0 be a solution of the equation
Z
def
min{l−2 kxk2 , l−1 kxk} Q(dx) = 1.
(3)
ξ(l) =
H

15

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Electronic Communications in Probability

It follows from (2) that l(Q) is uniquely defined.
Note that (3) only depends on the projection of Q on R+ , which we denote by Qr ,
i.e.
def
(4)
Qr ([t, ∞)) = Q({x ∈ H : kxk ≥ t}), t > 0.
Thus we can also write (3) in the form
Z ∞
min{l−2 x2 , l−1 x} Qr (dx)
ξ(l) =

(5)

0

and define l(Qr ) to be the value of l for which the integral in (5) is equal to one.
Clearly l(Q) = l(Qr ).
The following theorem is the main result of this paper.
Theorem 1.1 Under the above assumptions we have
(0.25)l(Qr ) ≤ EkXk ≤ (2.125)l(Qr ).

(6)

If Q is symmetric, the constant in the upper bound of (6) can be decreased to 1.25.
In preparation for the proof of this theorem we consider a decomposition of X. We
write X = Y + Z, where Y and Z are independent mean zero random vectors with
characteristic functions given by

E exp ihu, Y i = exp
E exp ihu, Zi = exp

Z

kxk 0 :

(42)

S

R
It is easy to see that X = S F dM can be described by (1) in which Q is the image
of the measure θ(dx, s)m(ds) under the map (s, x) → F (s)x. By the same argument
used in the proof of (37) we get
Z
(43)
(0.125)kF kΦ ≤ Ek F dM k ≤ (2.125)kF kΦ.
S

for every F satisfying (38).

Remark 2.1 Stochastic integrals in which the integrand is real valued and the random measure is Hilbert space valued and vice versa are special cases of (39), so that
(43) holds for them also.

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Electronic Communications in Probability

3

Examples

Let X, Q and Qr be as given in Section 1. In particular let


1
1
I
+
I
dx
Qr (dx) = C
xp+1 [0 0. A similar statement can be made when ψr is regularly varying at infinity
with index 1 < p < 2. In this case
lim

t→0

ζ(t)(p − 1)
= 1.
ψr (1/t)

(58)

Therefore, heuristicly, when ψr−1 (5(p − 1)/2) is very large
EkXk ≥

1−ǫ
4ψr−1 (5(p − 1)/2)

(59)

for all ǫ > 0. Both these statement can be made precise with specific examples.
Consider the stochastic integral in (34). Define
φ1 (u) = u2 I(u < 1) + uI(u ≥ 1),

u ≥ 0.

(60)

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Electronic Communications in Probability

Clearly
φ1 (u) ≤ φ0 (u) ≤ 2φ1 (u)
where φ0 is given in (31). Set
Z
def
e
φ1 (|hy, xi|) θ(dx, s),
φ(y, s) =
H

y ∈ H, s ∈ S.

(61)

(62)

It follows from Theorem 2.1 that

e e ≤ E|
(0.125)kfk
φ

Z

S

e e
hfe, dM i| ≤ (4.25)kfk
φ

(63)

where kfekφe is defined as in (32).
We proceed to evaluate kfekφe for M in (28) with θ(dx, s)m(ds) = θ(dr, dθ, s)m(ds) =
Cr−p−1 drσ(dθ)m(ds), similar to (49). A simple calculation shows that
kfekφe = inf{v > 0 : Dp

Z Z
S

∂U

|h

fe(s)
, θi|p dσ(θ) m(ds) ≤ 1}
v

(64)

where Dp = C/((2 − p)(p − 1)). Consequently

or, equivalently

kfekφe =

kfekφe =

 Z Z
Dp
S

∂U

1/p
|hf (s), θi| dσ(θ) m(ds)
p

 Z
Z
p
Dp
kf (s)k


1/p
f (s)
p
, θi| dσ(θ) m(ds)
|h
.
kf (s)k
∂U

S

(65)

(66)

We give two specific examples. Let fk (s) be the k–th coordinate of f (s) and ek =
(0, . . . , 1, . . .) be the unit vector in the k–th coordinate. Suppose that the measure σ
assigns mass pk /2 to both ek and −ek . Then it follows from (65) that
kfekφe =

Dp

∞ Z
X

k=1

S

 !1/p
.
|fk (s)| m(ds) pk
p

(67)

For the second example let σ be uniform measure of mass one on the unit sphere in
Rn . It follows from (66) that
kfekφe =



Dp E|g|p
n+p−1
E|g|
E|g|n−1

Z

S

1/p
kf (s)kp m(ds)

(68)

where g is a normal random variable with mean zero and variance one.
R
p
To derive (68) we calculate ∂U |h kff (s)
(s)k , θi| dσ(θ). Clearly this integral remains the
n
same if we rotate the unit sphere in R . Thus it remains the same for any unit vector
f (s)/kf (s)k in Rn . We take e1 for this vector.

L1 –norm of infinitely divisible random vectors

29

Let ξ = (ξ1 , . . . , ξn ) where {ξi }ni=1 are independent normal random variables with
mean zero and variance one. Note that
Z ∞
Z ∞
2
...
|he1 , ξi|p e−kξk /2 dξ
E|ξ1 |p = (2π)−n/2
Z∞∞
Z∞∞
ξ p −kξk2 /2
−n/2
i| e
dξ.
(69)
kξkp |he1 ,
...
= (2π)
kξk
∞
∞
Writing this in polar coordinates we get
Z ∞
Z
p
−n/2
p+n−1 −r 2 /2
|he1 , θi|p dθ
r
e
dr
E|ξ1 | = (2π)
∂U
0
Z
|he1 , θi|p dθ.
= 2−(n+1)/2 π −(n−1)/2 E|ξ1 |n+p−1

(70)

∂U

Consequently

Z

∂U

|he1 , θi|p dθ =

2(n+1)/2 π (n−1)/2 E|ξ1 |p
.
E|ξ1 |n+p−1

(71)

Taking p = 0 in (71) gives the surface area of the unit sphere in Rn . Dividing (71)
R
p
by this number gives ∂U |h kff (s)
(s)k , θi| dσ(θ) and hence (68).
It is not hard to obtain such definite results for integrals with respect to other infinitely
divisible random measures than stable ones. It only requires a good estimate for φe
in (62), which is not hard to obtain when the L´evy measure θ associated with M in
(28) has nice properties, such as being regularly varying at zero and infinity.

References
[1] V.H. de la Pe˜
na and E. Gin´e. Decoupling. From dependence to independence.
Springer–Verlag, 1999.
[2] N. Dunford and J.T. Schwartz. Linear Operators, Part I: General Theory. Wiley,
1958.
[3] I.I. Gihman and A.V. Skorohod. The Theory of Stochastic Processes, Vol. 2,
Springer, Berlin, 1975.
[4] M.J. Klass. Precision bounds for the relative error in the approximation of E|Sn |
and extensions. Ann. Probab., 8: 350–367, 1980.
[5] S. Kwapie´
n and W.A. Woyczy´
nski. Random Series and Stochastic Integrals: Single
and Multiple. Probability and its Applications Birkhauser, 1992.
[6] M. B. Marcus and J. Rosi´
nski. Continuity of stochastic integrals with respect to
infinitely divisible random measures. Preprint.
[7] K. R. Parthasarathy. Probability measures on metric spaces, Academic Press, New
York, 1967.