stable random variable

A real random variable $X$ defined on a probability space $(\mathrm{\Omega },ℱ,P)$ 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 $a,b\in ℝ$.

• 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 $a,b\in ℝ$, there exist $c,d\in ℝ$ 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	2013-03-22 16:25:56
Last modified on	2013-03-22 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