Limit distributions for sums of independent random vectors. Heavy tails in theory and practice (Q2719286)

In this comprehensive monograph which for the years to come can be expected to be the most important reference text for the treated subject the authors offer a detailed presentation of the theory of multivariate stable probability distributions in the framework of their concept of operator stable (or semistable) laws. So, they provide a valuable coherent text, useful for theoreticians as well as for practitioners, e.g. for researchers in anomalous diffusion processes. In their preface they describe (modestly) their motivation in the following words: `` `Limit distributions for sums of independent random variables' was published in Russian by B. V. Gnedenko and A. N. Kolmogorov in 1949 and translated into English by K. L. Chung in 1968. This book provided an accessible but serious introduction to the central limit theory of random variables, which lies at the heart of probabilty and statistics. It required only a knowledge of analysis, yet it took the reader to the frontiers of current research. Fifty years later, there is no better reference for much of this material. Just as important, the exposition has provided a framework for research on these fundamental principles. We set out to write another book in the same spirit, except that now the basic theory can be presented in a multivariable setting. There is an old adage in mathematics: The right proof is the proof that generalizes. History has validated the work of Gnedenko and Kolmogorov because their approach to the one variable extends to the multivariate case.'' NEWLINENEWLINENEWLINEWell, the authors of the present book deserve to be congratulated to the impressive result of their undertaking. Their book is divided into four parts. Part I (Introduction) is an outline of the required foundations (random vectors, linear operators, infinitely divisible laws, triangular arrays, domains of attraction). Part II is devoted to multivariate regular variation for linear operators, real valued functions and Borel measures. In the essential Part III the preceding tools are used to obtain ``multivariate limit theorems'' concerning operator semistable and stable laws, in particular ``central limit theorems'', normal and non-normal limits, statements on large deviations, law of the iterated logarithm. In the final Part IV applications to statistics and to self-similar stochastic processes are discussed. Most of the eleven chapters are rounded up by concluding ``Notes and Comments'', outlining the sources of results presented (so giving a useful survey of relevant literature) and hinting to open questions. In the References 199 titles are listed.