Vol. 91, No. 1, 1980

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On a characterization using random sums

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

Vol. 91 (1980), No. 1, 71–77

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 Xk, 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
Received: 20 December 1976
Revised: 30 May 1980
Published: 1 November 1980
J. R. Choike
Ignacy I. Kotlarski
V. M. Smith