Infinite divisibility (probability)

In probability theory, to say that a probability distribution F on the real line is infinitely divisible means that if X is any random variable whose distribution is F, then for every positive integer n there exist n independent identically distributed random variables X1, ..., Xn whose sum is equal in distribution to X (those n other random variables do not usually have the same probability distribution as X).

The Poisson distribution, the negative binomial distribution, and the Gamma distribution are examples of infinitely divisible distributions; as are the normal distribution, Cauchy distribution and all other members of the stable distribution family. The skew-normal distribution is an example of a non-infinitely divisible distribution (See Domínguez-Molina and Rocha Arteaga (2007))

Infinitely divisible distributions appear in a broad generalization of the central limit theorem: the limit of the sum of independent uniformly asymptotically negligible (u.a.n.) random variables within a triangular array approaches &mdash; in the weak sense &mdash; an infintely divisible distribution. The u.a.n. condition is given by


 * $$\lim_{n\to\infty} \max_k \; P( \left| X_{nk} \right| > \varepsilon ) = 0 \text{ for every }\varepsilon > 0.$$

Thus, for example, if the uniform asymptotic negligibility condition is satisfied via an appropriate scaling of identically distributed random variables with finite variance, the weak convergence is to the normal distribution in the classical version of the central limit theorem. More generally, if the u.a.n. condition is satisfied via a scaling of identically distributed random variables (with not necessarily finite second moment), then the weak convergence is to a stable distribution. On the other hand, for a triangular array of independent (unscaled) Bernoulli random variables where the u.a.n. condition is satisfied through


 * $$\lim_{n\rightarrow\infty} np_n = \lambda,$$

the weak convergence of the sum is to the Poisson distribution with mean λ as shown the familiar proof of the law of small numbers.

Every infinitely divisible probability distribution corresponds in a natural way to a Lévy process, i.e., a stochastic process { Xt : t ≥ 0 } with stationary independent increments (stationary means that for s < t, the probability distribution of Xt &minus; Xs depends only on t &minus; s; independent increments means that that difference is independent of the corresponding difference on any interval not overlapping with [s, t], and similarly for any finite number of intervals).

This concept of infinite divisibility of probability distributions was introduced in 1929 by Bruno de Finetti.

See also indecomposable distribution.

Relevance to statistics
The concepts of the decomposition of distributions and infinite divisibility arise in statistics in relationship to seeking families of distributions that, similarly to the normal distribution, might be expected to be appropriate in general circumstances. Given that the normal distribution arises in connection with the central limit theorem as, loosely speaking, the distribution of the average of an infinite number of identically distributed random variables, a closely related question is to ask which distributions can arise naturally as the sum of a finite number of identically distributed random variables.