Infinite Divisibility - in Probability Distributions

In Probability Distributions

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, ..., 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, the stuttering 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).)

Every infinitely divisible probability distribution corresponds in a natural way to a Lévy process, i.e., a stochastic process { X_t : t ≥ 0 } with stationary independent increments (stationary means that for s < t, the probability 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, and similarly for any finite number of intervals).

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