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Book review

Author(s) Dall'Aglio, G. (ed.)
Kotz, S. (ed.)
Solinetti, G. (ed.)
Title Advances in probability distributions with given marginals
Publisher Kluwer
Year of publication 1991
Reviewed by Günter Vojta

This book comprizes the proceedings of a special symposium on probability theory organized in 1990 by the Department of Statistics of the University La Sapienza at Rome. It was devoted to a very important modern topic, i.e. the theory of multivariate probability distributions, their determination by marginal distributions, and their applications, including wide fields such as probabilistic topological and metric spaces, semigroups, concepts of convolution and, generally, the interdependence of random variables.

The volume contains 11 invited lectures of varying length, partly reviews and partly original contributions given by well-known authors. The reports by G. Dall'Aglio and by B. Schweizer on the foundations and the historical development of the general theory give a very good introduction into the whole area. R.B. Nelsen and L. Rüschendorf present interesting surveys about bivariate and multivariate distributions, relationships between variates, inequalities and Fréchet bounds, and their applications. In the foreground stands the concept of copulas, a copula being a function which couples a multivariate distribution function to its marginal distribution functions. Other topics treated are partial orderings for distributions, measures of dependence in multivariate distributions, indecomposable marginal problems and extremal solutions in marginal problems. Works by M.J. Frank on the algebraic theory of generalized convolutions for dependent random variables and by S. Cambanis on random processes with correlations are of particular interest.

The papers are supplemented by often long lists of valuable references with titles. An author index and a subject index are added. The papers are reproduced from different original typescripts.

The publication of this volume is highly to appreciate. It renders a huge amount of professional information available to a wide audience of experts, research workers and students of probability theory and its applications in many fields. For libraries it is certainly a must.