On generalized gamma and generalized negative binomial convolutions. Part I

Abstract

1. Introduction and summary

In order to prove the infinite divisibility (inf. div.) of the lognormal distribution, O. Thorin introduced the generalized gamma convolutions (g.g.c.) and developed their theory in a series of papers. The lognormal distribution turned out to be a g.g.c. and hence inf. div. Thorin's significant first papers (Thorin, 1977 a, b) were rapidly followed by others by different authors; see e.g. the references in Bondesson (1977). Using Thorin's approach, the present author (Bondesson, 1977) obtained a simple set of sufficient conditions for a distribution on [0, ∞) to be a g.g.c. which seems to be the most general one that exists at present. Throughout the present paper the reader is assumed to be familiar with the g.g.c.'s. A short review is given in e.g. Bondesson (1977).