On generalized gamma and generalized negative binomial convolutions. Part II

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 a_i 's and they γ_i '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 p_j =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).