Stable distribution

A probability distribution is stable when every linear combination of independent copies has the same distribution up to a location and scale change. The Cauchy distribution is stable with stability parameter one.

A probability distribution is strictly stable with index $\alpha$ if $n^{-1/\alpha }(T^{(1)}+\cdots +T^{(n)})\stackrel{d}{=}T$ for every positive integer $n$, for independent copies. Here no centering term is allowed, unlike in the general definition of a stable distribution.