infinitely divisible random variable
Let $n$ be a positive integer. A real random variable^{} $X$ defined on a probability space^{} $(\mathrm{\Omega},\mathcal{F},P)$ is said to be

1.
$n$decomposable if there exist $n$ independent random variables ${X}_{1},\mathrm{\dots},{X}_{n}$ such that $X$ is identically distributed as the sum ${X}_{1}+\mathrm{\cdots}+{X}_{n}$. A $2$decomposable random variable is also called a decomposable random variable;

2.
$n$divisible if $X$ is $n$decomposable and the ${X}_{i}$’s can be chosen so that they are identically distributed;

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 function^{} 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 distributions^{}, negative binomial distributions^{}, and compound Poisson distributions.
Title  infinitely divisible random variable 

Canonical name  InfinitelyDivisibleRandomVariable 
Date of creation  20130322 16:25:58 
Last modified on  20130322 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 