Infinitely-divisible distributions, factorization of
A representation of infinitely-divisible distributions in the form
of the convolution of certain probability distributions. The
distributions which participate in the factorization
of infinitely-divisible distributions are called the
components in the factorization.
Certain factorizations of infinitely-divisible distributions may have
components which are not infinitely divisible
[1].
An important task in the theory of factorization of
infinitely-divisible distributions is the description of the class
of infinitely-divisible distributions with exclusively
infinitely-divisible components. The representatives of
include the
normal distribution,
the
Poisson distribution
and their compositions (cf.
Lévy–Cramér theorem).
An important role in the description of the class
is played by Linnik's class
of infinitely-divisible distributions
[2],
in which the function
in the Lévy–Khinchin canonical representation is a step
function with jumps at the points between
 ,
where
 ,
 ,
and the numbers
( ;
 )
are natural numbers other than 1. If the infinitely-divisible distribution is such that
 ,
it can only belong to
if it belongs to
 .
This condition is not sufficient, but it is known that a distribution of
belongs to
if
for some
and
 .
If
 ,
belonging to
is not a necessary condition for belonging to
 .
For instance, all infinitely-divisible distributions in which the function
is constant for
and
 ,
where
 ,
belong to
 .
The following is a simple sufficient condition for
an infinitely-divisible distribution not to belong to
 .
The inequality
must be fulfilled on the interval
 ,
where
 .
It follows from this condition that a
stable distribution,
except the normal distribution and the unit distribution,
as well as the gamma-distribution and the
 -distribution,
does not belong to
 .
The class
is dense in the class of all infinitely-divisible distributions
in the topology of weak convergence; all infinitely-divisible distributions can
be represented as compositions of a finite or countable set of distributions from
 .
References| [1] |
A.Ya. Khinchin,
"Contribution à l'arithmétique des lois de distribution"
Byull. Moskov. Gos. Univ. (A)
, 1
: 1
(1937)
pp. 6–17 | | [2] |
Yu.V. Linnik,
"General theorems on factorization of infinitely divisible laws"
Theory Probab. Appl.
, 3
: 1
(1958)
pp. 1–37
Teor. Veroyatnost. i Primenen.
, 3
: 1
(1958)
pp. 3–40 | | [3] |
Yu.V. Linnik,
"Decomposition of probability laws"
, Oliver & Boyd
(1964)
(Translated from Russian) | | [4] |
Yu.V. Linnik,
I.V. Ostrovskii,
"Decomposition of random variables and vectors"
, Amer. Math. Soc.
(1977)
(Translated from Russian) | | [5] |
B. Ramachandran,
"Advanced theory of characteristic functions"
, Statist. Publ. Soc.
, Calcutta
(1967) | | [6] |
E. Lukacs,
"Characteristic functions"
, Griffin
(1970) | | [7] |
L.Z. Livshits,
I.V. Ostrovskii,
G.P. Chistyakov,
"Arithmetic of probability laws"
J. Soviet Math.
, 6
: 2
(1976)
pp. 99–122
Itogi Nauk. i Tekhn. Teor. Veroyatnost. Mat. Statist. Teoret. Kibernetika
, 12
(1975)
pp. 5–42 | | [8] |
I.V. Ostrovskii,
"The arithmetic of probability distributions"
Theor. Probab. Appl.
, 31
: 1
(1987)
pp. 1–24
Teor. Veroyatnost. i Primenen.
, 31
: 1
(1986)
pp. 3–30 |
I.V. Ostrovskii
CommentsReferences| [a1] |
E. Lukacs,
"Developments in characteristic function theory"
, Griffin
(1983) |
This text originally appeared in Encyclopaedia of Mathematics
- ISBN 1402006098
|