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

1. $X$ is not trivial; that is, the range of the distribution function of $X$ strictly includes $\lbrace 0,1\rbrace$ and
2. given any positive integer $n$ and $X_1,\ldots,X_n$ random variables, iid as $X$ $$S_n:=X_1+\cdots +X_n \stackrel{t}{=} X.$$ 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 $aX_1+bX_2$ and $cX+d$ are identically distributed.
• A stable distribution function is absolutely continuous and infinitely divisible.

Some common stable distribution functions are the normal distributions and Cauchy distributions.