Generalized gamma convolutions and related classes of distributions and densities
type: Item
instance of: scholarly article
title: Generalized gamma convolutions and related classes of distributions and densities
zbMATH Open document ID: 0756.60015
author: Lennart Bondesson
published in: Lecture Notes in Statistics
publication date: 1992-09-18T00:00:00Z
review text: This is a book on infinite divisibility, though this term does not appear in the title. It is the first look on this subject, or rather on special part of it. The reviewer [Stochastic Processes Appl. 1, 125-143 (1973; Zbl 0259.60011)] posed a number of questions, all of which seem to have been answered by now. One of these questions concerned the possible infinite divisibility of the log-normal distribution, which was settled in the affirmative by \textit{O. Thorin} [Scand. Actuarial J. 1977, 121-148 (1977; Zbl 0372.60020)], to whom this book has been dedicated. Thorin's proof was long and technical and involved the definition of the main class of distributions studied in this book: the generalized gamma convolutions (GGC's). These are essentially convolutions of gamma distributions, possibly shifted, and their limits. The GGC's, previously studied by the author in a long series of articles, turn out to be quite a large class, all elements of which are, trivially, infinitely divisibile. The difficulty is to show that certain distributions such as the log-normal belong to this class, which is denoted by \({\mathcal T}\) in the book, in honor of Thorin. The class \({\mathcal T}\) is closely connected to a very curious and only partly understood class of functions introduced by Bondesson, named hyperbolically completely monotone (HCM), and defined by the requirement (on \(f\)) that \(f(uv)f(u/v)\) is completely monotone as a function of \(v+v^{-1}\) for every \(u>0\). The HCM's have several interesting properties, among which simple closure properties, which are quite useful in combination with the property that a function \(\varphi\), with \(\varphi(0)=1\), is the Laplace transform of a GGC if and only if it is HCC. The material described above constitutes the core of the book, about two thirds. The remaining chapters are devoted to generalizations and to the analogue of GGC's for distributions on \(\mathbb{Z}_ +\), the generalized negative binomial convolutions (GNBC…
Mathematics Subject Classification ID: 60E07
Mathematics Subject Classification ID: 60E10
Mathematics Subject Classification ID: 30E20
Mathematics Subject Classification ID: 60-02
zbMATH DE Number: 54300
zbMATH Keywords: infinite divisibility
zbMATH Keywords: generalized gamma convolutions
zbMATH Keywords: completely monotone
zbMATH Keywords: Laplace transform
reviewed by: Fred W. Steutel
MaRDI profile type: Publication
Recommended article: On generalized gamma convolution distributions
Recommended article: Q5462011
Recommended article: Q3563036
Recommended article: On univariate and bivariate generalized gamma convolutions
Recommended article: Q4582877
Recommended article: An approximation to the convolution of gamma distributions
Recommended article: Q5411322
Recommended article: A Review on Convolutions of Gamma Random Variables
Recommended article: On mixtures of the normal distribution by the generalized gamma convolutions
Recommended article: Density and distribution evaluation for convolution of independent gamma variables
description: scientific article; zbMATH DE number 54300