Editing Infinite divisibility (section)

==In probability distributions==

{{Main|infinite divisibility (probability)}}

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'' [[statistical independence|independent]] [[identically distributed]] random variables ''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub> 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 stuttering Poisson distribution,{{citation needed|date=September 2019}} 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|skew-normal distribution]] is an example of a non-infinitely divisible distribution. (See Domínguez-Molina and Rocha-Arteaga (2007).)

Every infinitely divisible probability distribution corresponds in a natural way to a [[Lévy process]], i.e., a [[stochastic process]] { ''X<sub>t</sub>'' : ''t'' ≥ 0 } with stationary independent increments (''stationary'' means that for ''s'' < ''t'', the [[probability distribution]] of ''X''<sub>''t''</sub> &minus; ''X''<sub>''s''</sub> depends only on ''t'' &minus; ''s''; ''independent increments'' means that that difference is [[statistical independence|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]].