Characterizations of discrete distributions using the Rao-Rubin condition

Abstract

Consider the multivariate splitting model N = N1 + ⋯ + Nk, where N1,..., Nk, k ≥ 3, are arbitrary (not necessarily independent) random variables (r.v.'s) taking values in ℕ = {0, 1,...}, and assume that the Rao-Rubin condition is satisfied for N1 and N2. Also assume that the conditional distribution of the vector (N1,..., Nk) given N is a convolution type. Characterizations related to this model (with k = 2) was first considered by Shanbhag (1977. J. Appl. Probab. 14, 640-646), as an extension of the binomial damage model established by Rao and Rubin (1964. Sankhyā Ser. A 26, 295-298), and was extended to any k ≥ 3 by Rao and Srivastava (1979. Sankhyā Ser. A 41, 124-128). In the present paper we provide an alternative set of conditions, under which the distribution of N is characterized, and we apply the result to some discrete distributions. © 2005 Elsevier B.V. All rights reserved.