On generalized gamma and generalized negative binomial convolutions. Part II

Abstract

Abstract 1. Introduction and summary Part II of this paper is the result of the present author's efforts to show that a distribution on Z +={0, 1, 2, ... } with probability function of the form where the ai 's and they γi 's are strictly positive and , is a generalized negative binomial convolution (g.n.b.c.). The g.n.b.c.'s were introduced in Part I of this paper. It will be shown that a distribution of the above type at least for N = 1 is a g.n.b.c. In particular the discrete Pareto distribution pj =constant·(j+a)-γ,j=0, 1, 2, ..., (a>0, γ>1) is a g.n.b.c. The proof will be based on a result for generalized gamma convolutions (g.g.c.) which is equivalent to Theorem 2 in Bondesson (1977).