# infinitely divisible random variable

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

1. 1.

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

2. 2.

$n$-divisible if $X$ is $n$-decomposable and the $X_{i}$’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 $n$.

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 InfinitelyDivisibleRandomVariable 2013-03-22 16:25:58 2013-03-22 16:25:58 CWoo (3771) CWoo (3771) 8 CWoo (3771) Definition msc 60E07 $n$-decomposable $n$-divisible infinitely divisible distribution infinitely divisible decomposable random variable