Random Stability of Random Variables

For a random variable $N=0,1,2,\dots$ we study the following question: When does the sum of $N$ many independent and identically distributed copies of a random variable $X$ have the same law a a nontrivial rescaling of $X$? We show that such $N$-stable random variable exists if and only $1<\mathbb{E}[N]<\mathrm{\infty }$. Under an additional assumption $\mathbb{E}[N\ln N]<\mathrm{\infty }$, we describe all $N$-stable $X$. We also study a converse problem: For a given $X\ge 0$ with $\mathbb{E}[X]=1$, we study the set of all $N$ such that $X$ is $N$-stable. Distributions of $N$ form a semigroup with respect to composition of probability generating functions. We show these probability generating functions need to commute with respect to composition. We present explicit families of composition semigroups. Equivalent formulations have appeared in difference forms, and this article aims to unify and extend them.