- Infinite divisibility (probability)
In
probability theory , to say that aprobability distribution "F" on the real line is infinitely divisible means that if "X" is anyrandom variable whose distribution is "F", then for every positive integer "n" there exist "n" independentidentically distributed random variables "X"1, ..., "X""n" 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 , thenegative binomial distribution , and theGamma distribution are examples of infinitely divisible distributions; as are thenormal distribution ,Cauchy distribution and all other members of thestable distribution family.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 — in the weak sense — an infintely divisible distribution. The u.a.n. condition is given by:
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 thenormal 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:
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., astochastic process { "Xt" : "t" ≥ 0 } with stationary independent increments ("stationary" means that for "s" < "t", theprobability distribution of "X""t" − "X""s" depends only on "t" − "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 thenormal distribution , might be expected to be appropriate in general circumstances. Given that thenormal distribution arises in connection with thecentral 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.References
* Domínguez Molina, J.A. y Rocha Arteaga, A. "On the Infinite Divisibility of some Skewed Symmetric Distributions". "Statistics and Probability Letters", V. 77, Issue 6, 644–648, 2007.
* Steutel, F. W. (1979), "Infinite Divisibility in Theory and Practice" (with discussion), "Scandinavian Journal of Statistics." 6, 57–64.
* Steutel, F. W. and Van Harn, K. (2003), "Infinite Divisibility of Probability Distributions on the Real Line" (Marcel Dekker).
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