infinitely divisible random variable

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