A probability distribution which, for any
may be represented as a composition (convolution) of
identical probability distributions. The definition of an infinitely-divisible distribution is
applicable to an equal degree to a distribution
on the straight line, on a finite-dimensional Euclidean space
and to a number of other, even more general,
cases. The one-dimensional case will be considered below.
The characteristic function
of an infinitely-divisible distribution is called
infinitely divisible.
Such a function may be represented, for any value of
,
as the
-th
power of some other characteristic function:
Examples of infinitely-divisible distributions include the
normal distribution,
the
Poisson distribution,
the
Cauchy distribution,
and the
"chi-squared" distribution.
The property of infinite divisibility is most easily
tested by using characteristic functions. The composition of
infinitely-divisible distributions and the limit of weakly-convergent
sequences of infinitely-divisible distributions are again infinitely divisible.
A random variable, defined on some probability space, is called
infinitely divisible
if it can be represented, for any
,
as a sum of
independent identically-distributed random variables defined on
that space. The distribution of each such
variable is infinitely divisible, but the converse is not
always true. Consider, e.g., the discrete probability space formed by
with Poisson probabilities
The random variable
is not infinitely divisible, even though
its probability distribution (Poisson distribution) is infinitely divisible.
Infinitely-divisible distributions first appeared in connection with
the study of stochastically-continuous homogeneous stochastic
processes with stationary independent increments (cf.
Stochastic process with stationary increments;
Stochastic process with independent increments)
[1],
[2],
[3].
This is the name of processes
,
,
which satisfy the following requirements: 1)
;
2) the probability distribution of the increment
,
,
depends only on
;
3) for
the differences
are mutually-independent random variables; 4) for any
,
as
.
For such a process the value
for any
will be an infinitely-divisible random variable, and
the corresponding characteristic function satisfies the relation
The general form of
for such processes — on the assumption that the variances
are finite
— was found by
A.N. Kolmogorov
[2]
(a special case of the canonical
representation of infinitely-divisible distributions presented below).
The characteristic function of an infinitely-divisible distribution never vanishes,
and its logarithm (in the sense of the
principal value) permits a representation of the form:
(the so-called
Lévy–Khinchin canonical representation),
where
is some real constant and
is a non-decreasing function of bounded variation with
.
The integrand is taken to be equal to
for
.
Whatever the value of the constant
and of the function
with the above properties, formula
(*)
defines
the logarithm of the characteristic function
of some infinitely-divisible distribution. The correspondence
between infinitely-divisible distributions and pairs
is one-to-one and is also
bicontinuous.
This means that an infinitely-divisible distribution is
weakly convergent towards an infinitely-divisible limit distribution if and only if
and
converges to
as
.
Examples.
Let
,
,
,
.
Then, in order to have a normal distribution with mathematical expectation
and variance
in formula
(*),
one must put
For a Poisson distribution with parameter
one has
For a Cauchy distribution with density
one has
,
The canonical representation
(*)
is convenient from a purely
"technical"
point of view (owing to the fact that
has bounded variation), but the function
has no direct probabilistic interpretation. For this
reason another form of representation of infinitely-divisible distributions,
which permits a direct probabilistic interpretation, is
employed as well. Let the functions
and
be defined, for
and
respectively, by the formulas:
These functions are non-decreasing,
for
,
and
for
;
in a neighbourhood of zero the functions may be unbounded. If one denotes by
the jump of
at zero, formula
(*)
may be rewritten as follows:
(Lévy's canonical representation).
The functions
and
describe, roughly speaking, the frequency of the jumps
of varying quantities in the homogeneous process
with independent increments for which
The importance of the role played in the limit theorems
of probability theory by infinitely-divisible distributions is due
to the fact that these and only these distributions
can be the limit distributions for sums of
independent random variables subject to the requirement of
asymptotic negligibility.
Consider the triangular array
,
of mutually-independent random variables and select mutually-independent random variables
with infinitely-divisible distributions (the so-called
accompanying infinitely-divisible distributions);
the characteristic function
of the variable
is defined in terms of the characteristic function
of the variable
so as to preserve the following basic property: The distributions of the sums
converge to the same limit distribution (for a certain selection of the constants
)
if and only if the sums
converge to a limit distribution. For a symmetric distribution
it is assumed that
In other cases the expression for
is more complex, and contains the so-called truncated mathematical expectations of
.
The properties of infinitely-divisible distributions are described in
terms of functions forming part of the
canonical representations. For instance, an infinitely-divisible distribution function
is continuous if and only if
.
An important special case of infinitely-divisible distributions
are the so-called stable distributions (cf.
Stable distribution).
See also
Infinitely-divisible distributions, factorization of.