# stable random variable

A real random variable  $X$ defined on a probability space  $(\Omega,\mathcal{F},P)$ is said to be stable if

1. 1.

$X$ is not trivial; that is, the range of the distribution function  of $X$ strictly includes $\{0,1\}$, and

2. 2.

given any positive integer $n$ and $X_{1},\ldots,X_{n}$ random variables, iid as $X$:

 $S_{n}:=X_{1}+\cdots+X_{n}\lx@stackrel{{t}}{{=}}X.$

In other words, there are real constants $a,b$ such that $S_{n}$ and $aX+b$ have the same distribution functions; $S_{n}$ and $X$ are of the same type.

Furthermore, $X$ is strictly stable if $X$ is stable and the $b$ given above can always be take as $0$. In other words, $X$ is strictly stable if $S_{n}$ and $X$ belong to the same scale family.

A distribution function is said to be stable (strictly stable) if it is the distribution function of a stable (strictly stable) random variable.

Remarks.

Title stable random variable StableRandomVariable 2013-03-22 16:25:56 2013-03-22 16:25:56 CWoo (3771) CWoo (3771) 13 CWoo (3771) Definition msc 60E07 stable distribution function strictly stable random variable strictly stable distribution function