infinitely divisible random variable

Let n be a positive integer. A real random variableMathworldPlanetmath X defined on a probability spaceMathworldPlanetmath (Ω,,P) is said to be

  1. 1.

    n-decomposable if there exist n independent random variables X1,,Xn such that X is identically distributed as the sum X1++Xn. A 2-decomposable random variable is also called a decomposable random variable;

  2. 2.

    n-divisible if X is n-decomposable and the Xi’s can be chosen so that they are identically distributed;

  3. 3.

    infinitely divisible if X is n-divisible for every positive integer n. In other words, X can be written as the sum of n iid random variables for any n.

A distribution functionMathworldPlanetmath is said to be infinitely divisible if it is the distribution function of an infinitely divisible random variable.

Remark. Any stable random variable is infinitely divisible.

Some examples of infinitely divisible distribution functions, besides those that are stable, are the gamma distributionsMathworldPlanetmath, negative binomial distributionsMathworldPlanetmath, and compound Poisson distributions.

Title infinitely divisible random variable
Canonical name InfinitelyDivisibleRandomVariable
Date of creation 2013-03-22 16:25:58
Last modified on 2013-03-22 16:25:58
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 8
Author CWoo (3771)
Entry type Definition
Classification msc 60E07
Defines n-decomposable
Defines n-divisible
Defines infinitely divisible distribution
Defines infinitely divisible
Defines decomposable random variable