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The characteristic function of any infinitely divisible distribution is then called an infinitely divisible characteristic function. [ 1 ] More rigorously, the probability distribution F is infinitely divisible if, for every positive integer n , there exist n i.i.d. random variables X n 1 , ..., X nn whose sum S n = X n 1 + ... + X nn has the ...
Infinite divisibility arises in different ways in philosophy, physics, economics, order theory (a branch of mathematics), and probability theory (also a branch of mathematics). One may speak of infinite divisibility, or the lack thereof, of matter , space , time , money , or abstract mathematical objects such as the continuum .
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 infinitely divisible probability distribution. It is also a strictly stable distribution. [8] Like all stable distributions, the location-scale family to which the Cauchy distribution belongs is closed under linear transformations with real coefficients.
Every infinitely divisible probability distribution is a limit of compound Poisson distributions. [1] And compound Poisson distributions is infinitely divisible by the definition. Discrete compound Poisson distribution
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 .
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].
The log-Cauchy distribution is infinitely divisible for some parameters but not for others. [10] Like the lognormal distribution, log-t or log-Student distribution and Weibull distribution, the log-Cauchy distribution is a special case of the generalized beta distribution of the second kind.