# 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 $X$:

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

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 $aX_{1}+bX_{2}$ 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 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