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.