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Homestable random variable
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stable random variable
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 $\{0,1\}$, and
2.
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.
Some common stable distribution functions are the normal distributions and Cauchy distributions.
Mathematics Subject Classification
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