On a characterization using random sums

J. R. Choike, Ignacy I. Kotlarski and V. M. Smith

Vol. 91 (1980), No. 1, 71–77

DOI: 10.2140/pjm.1980.91.71

Abstract

Let X₁, X₂, and X₃ be independent random variables and let Z₁ = X₁ + X₃ and Z₂ = X₂ + X₃. It is known that if the characteristic functions of X_k, k = 1,2,3, do not vanish then the distribution of (Z₁,Z₂) determines the distributions of X₁, X₂, and X₃ up to a shift. The aim of this paper is to prove a result of a similar nature using sums of a random number of random variables. We shall use ∼ for “has the same distribution as,” r.v. for “random variable,” ch.f. for “characteristic function,” and p.g.f. for “probability generating function.”

Mathematical Subject Classification 2000

Primary: 60E05

Secondary: 62E10

Milestones

Received: 20 December 1976

Revised: 30 May 1980

Published: 1 November 1980