Random stability of random variables

Andrey Sarantsev
Theory of Stochastic Processes
Vol.30 (46), no.1, 2026, pp.85-103

For a random variable N = 0, 1, 2, ... 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 as a nontrivial rescaling of X? We show that such N-stable random variable exists if and only 1 < E[N] < ∞. Under an additional assumption E[N ln N] < ∞, we describe all N-stable X. We also study a converse problem: For a given X ≥ 0 with E[X] = 1, find the set of all N such that X is N-stable. Distributions of these N form a commuting semigroup with respect to composition of probability generating functions.

DOI: https://doi.org/10.3842/tsp-4433942009-80
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