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More rigorously, the probability distribution F is infinitely divisible if, for every positive integer n, there exist n i.i.d. random variables X n1, ..., X nn whose sum S n = X n1 + ... + X nn has the same distribution F. The concept of infinite divisibility of probability distributions was introduced in 1929 by Bruno de Finetti.
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 ...
Pages in category "Infinitely divisible probability distributions" The following 18 pages are in this category, out of 18 total. This list may not reflect recent changes .
The Cauchy distribution is an example of a distribution which has no mean, variance or higher moments defined. Its mode and median are well defined and are both equal to . The Cauchy distribution is an infinitely divisible probability distribution. It is also a strictly stable distribution. [8]
Stable distributions are infinitely divisible. Stable distributions are leptokurtotic and heavy-tailed distributions, with the exception of the normal distribution (=). Stable distributions are closed under convolution. Stable distributions are closed under convolution for a fixed value of .
The distribution of a Lévy process has the property of infinite divisibility: given any integer n, the law of a Lévy process at time t can be represented as the law of the sum of n independent random variables, which are precisely the increments of the Lévy process over time intervals of length t/n, which are independent and identically ...
Infinitely divisible probability distributions (1 C, 18 P) L. ... Probability distributions with non-finite variance (30 P) S. Stable distributions (1 C, 8 P)
All infinitely divisible distributions are a fortiori decomposable; in particular, this includes the stable distributions, such as the normal distribution.; The uniform distribution on the interval [0, 1] is decomposable, since it is the sum of the Bernoulli variable that assumes 0 or 1/2 with equal probabilities and the uniform distribution on [0, 1/2].