The concepts of infinite divisibility and the decomposition of distributions arise in probability and statistics in relation to seeking families of probability distributions that might be a natural choice in certain applications, in the same way that the normal distribution is. The distributions sought correspond to random variables which are equivalent to the sums of a number of independent and identically distributed random variables, where the number of such variables can be set to any pre-specified number.
The term infinitely divisible characteristic function is used for the characteristic function of any infinitely divisible distribution.[1]
The concept of infinite divisibility of probability distributions was introduced in 1929 by Bruno de Finetti. These distributions play a very important role in probability theory in the context of limit theorems.[1]
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Definition
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).
Examples
The Poisson distribution, the negative binomial distribution, the exponential distribution, the geometric distribution, the Gamma distribution and the degenerate distribution are examples of infinitely divisible distributions; as are the normal distribution, Cauchy distribution and all other members of the stable distribution family. The uniform distribution and the binomial distribution are not infinitely divisible.[2]
Infinite divisibility in probability theory
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 infinitely divisible distribution. The u.a.n. condition is given by
Thus, for example, if the uniform asymptotic negligibility (u.a.n.) 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
the weak convergence of the sum is to the Poisson distribution with mean λ as shown by 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 − Xs 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).
See also
References
- ^ a b Lukacs, E. (1970) Characteristic Functions, Griffin , London. p. 107
- ^ Sato, Ken-iti (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press. p. 31. ISBN 978-0521553025.
- 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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