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

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

2.
given any positive integer $n$ and ${X}_{1},\mathrm{\dots},{X}_{n}$ random variables, iid as $X$:
$${S}_{n}:={X}_{1}+\mathrm{\cdots}+{X}_{n}\stackrel{t}{=}X.$$ 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.

•
If $X$ is stable, then $aX+b$ is stable for any $a,b\in \mathbb{R}$.

•
If $X$ and $Y$ are independent^{}, stable, and of the same type, then $X+Y$ is stable.

•
$X$ is stable iff for any independent ${X}_{1},{X}_{2}$, identically distributed as $X$, and any $a,b\in \mathbb{R}$, there exist $c,d\in \mathbb{R}$ such that $a{X}_{1}+b{X}_{2}$ and $cX+d$ are identically distributed.

•
A stable distribution function is absolutely continuous^{} (http://planetmath.org/AbsolutelyContinuousFunction2) and infinitely divisible.
Some common stable distribution functions are the normal distributions^{} and Cauchy distributions^{}.
Title  stable random variable 

Canonical name  StableRandomVariable 
Date of creation  20130322 16:25:56 
Last modified on  20130322 16:25:56 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  13 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 60E07 
Defines  stable distribution function 
Defines  strictly stable random variable 
Defines  strictly stable distribution function 