Infinite divisibility (probability)

In probability theory, a probability distribution is infinitely divisible if it can be expressed as the probability distribution of the sum of an arbitrary number of independent and identically distributed (i.i.d.) random variables. The characteristic function of any infinitely divisible distribution is then called an infinitely divisible characteristic function. More rigorously, the probability distribution F is infinitely divisible if, for every positive integer n, there exist n i.i.d. random variables Xn1, ..., Xnn whose sum Sn = Xn1 + ... + Xnn has the same distribution F. The concept of infinite divisibility of probability distributions was introduced in 1929 by Bruno de Finetti. This type of decomposition of a distribution is used in probability and statistics to find families of probability distributions that might be natural choices for certain models or applications. Infinitely divisible distributions play an important role in probability theory in the context of limit theorems. Examples of continuous distributions that are infinitely divisible are the normal distribution, the Cauchy distribution, the Lévy distribution, and all other members of the stable distribution family, as well as the Gamma distribution, the chi-square distribution, the Wald distribution, the Log-normal distribution and the Student's t-distribution. Among the discrete distributions, examples are the Poisson distribution and the negative binomial distribution (and hence the geometric distribution also). The one-point distribution whose only possible outcome is 0 is also (trivially) infinitely divisible. The uniform distribution and the binomial distribution are not infinitely divisible, nor are any other distributions with bounded support (≈ finite-sized domain), other than the one-point distribution mentioned above. The distribution of the reciprocal of a random variable having a Student's t-distribution is also not infinitely divisible. Any compound Poisson distribution is infinitely divisible; this follows immediately from the definition.