Random Stability of Random Variables

TL;DR

This paper investigates conditions under which a random variable remains stable under random sums, characterizing the distributions of the number of summands and their properties.

Contribution

It provides a complete characterization of N-stable random variables and the structure of their associated probability generating functions.

Findings

N-stable variables exist if and only if 1 < E[N] < ∞.

All N-stable variables are described under the condition E[N ln N] < ∞.

Probability generating functions of N form a commutative semigroup under composition.

Abstract

For a random variable $N = 0, 1, 2, \ldots$ 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 $1 < \mathbb E[N] < \infty$. Under an additional assumption $\mathbb E[N\ln N] < \infty$, 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.