Difference between revisions of "Infinitely-divisible distribution"
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A probability distribution which, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050910/i0509101.png" /> may be represented as a composition (convolution) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050910/i0509102.png" /> 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. | A probability distribution which, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050910/i0509101.png" /> may be represented as a composition (convolution) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050910/i0509102.png" /> 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. | ||
Revision as of 19:55, 27 February 2012
2020 Mathematics Subject Classification: Primary: 60E07 [MSN][ZBL]
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:
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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
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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
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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
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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
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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
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For a Poisson distribution with parameter one has
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For a Cauchy distribution with density
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one has ,
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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:
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![]() |
![]() |
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.
References
[1] | D. de Finetti, "Sulla funzione a incremento aleatorio" Atti Accad. Naz. Lincei Sez. 1 (6) , 10 (1929) pp. 163–168 |
[2] | A.N. Kolmogorov, "Ancora sulla forma generale di un processo stocastico omogeneo" Atti Accad. Naz. Lincei Sez. 1 (6) , 15 (1932) pp. 866–869 |
[3] | P. Lévy, "Sur les intégrales dont les éléments sont des variables aléatoires indépendantes" Ann. Scuola Norm. Sup. Pisa Ser. 2 , 3 (1934) pp. 337–366 |
[4] | P. Lévy, "Théorie de l'addition des variables aléatoires" , Gauthier-Villars (1937) |
[5] | A.Ya. Khinchin, "Limit laws for sums of independent random variables" , Moscow-Leningrad (1938) (In Russian) |
[6] | B.V. Gnedenko, A.N. Kolmogorov, "Limit distributions for sums of independent random variables" , Addison-Wesley (1954) (Translated from Russian) |
[7] | M. Fisz, "Infinitely divisible distributions: recent results and applications" Ann. of Math. Statist. , 33 (1962) pp. 68–84 |
[8] | V.V. Petrov, "Sums of independent random variables" , Springer (1975) (Translated from Russian) |
[9] | V.V. Sazonov, V.N. Tutubalin, "Probability distributions on topological groups" Theory Probab. Appl. , 11 : 1 (1966) pp. 1–45 Teor. Veroyatnost. i Primenen. , 11 : 1 (1966) pp. 3–55 |
Comments
References
[a1] | F.W. Steutel, "Infinite divisibility in theory and practice" Scand. J. Statist. , 6 (1979) pp. 57–64 |
[a2] | W. Feller, "An introduction to probability theory and its applications" , 1–2 , Wiley (1968–1971) |
Infinitely-divisible distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Infinitely-divisible_distribution&oldid=18038