Generalized gamma convolutions and related classes of distributions and densities
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's). The analysis of these distributions is less complete mainly because there seems to be no discrete analogue of the HCM's. The mathematics consists of analysis; Laplace transformation, complex function theory, some measure theory and only fairly elementary probability. It is an interesting book, well written, with many, sometimes surprising, examples and containing 152 references, almost half of them concerning infinite divisibility. Surprisingly, the word log- normal does not appear in the index or in the table of contents.
scientific article; zbMATH DE number 54300