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Laws of small numbers: Extremes and rare events. Disk. incl. (English) Zbl 0817.60057
DMV Seminar. 23. Basel: Birkhäuser,. xi, 282 p. (1994).
From the preface and the summary: This book is based on lectures given at the DMV Seminar on “Laws of small numbers: Extremes and rare events”, held at the Katholische Universität Eichstätt from October 20-27, 1991. In the first part of this book we will develop a theory of rare events for which a handy name is functional laws of small numbers. Whenever one is concerned with rare events, events with a small probability of occurrence, the Poisson distribution shows up in a natural way. It is reasonable to describe such events as truncated empirical point processes, and a point process of rare events can be approximated by a Poisson process. The second part of the book concentrates on the theory of extremes and other rare events of non i.i.d. random sequences. The rare events related to stationary sequences and independent sequences are considered as special cases of this general setup. The theory is presented in terms of extremes of random sequences as well as general triangular arrays of rare events.
Part I of this book is organized as follows: In Chapter 1 the general idea of functional laws of small numbers is made rigorous. Chapter 2 provides basic elements from univariate extreme value theory, which enable particularly the investigation of the peaks over threshold method as an example of a functional law of small numbers. In Chapter 3 we demonstrate how our approach can be applied to regression analysis or, generally, to conditional problems. Chapter 4 contains basic results from multivariate extreme value theory including their extension to the continuous time setting. The multivariate peaks over threshold approach is studied in Chapter 5. Chapter 6 provides some elements of exploratory data analysis for univariate extremes.
Part II considers non i.i.d. random sequences and rare events. Chapter 7 introduces the basic ideas to deal with the extremes and rare events in this case. These ideas are made rigorous in Chapter 8 presenting the general theory of extremes which is applied to the special cases of stationary and independent sequences. The extremes of nonstationary Gaussian processes are investigated in Chapter 9. Results for locally stationary Gaussian processes are applied to empirical characteristic functions. The theory of general triangular arrays of rare events is presented in Chapter 10, where we also treat general rare events of random sequences and the characterization of the point process of exceedances. This general approach provides a neat unification of the theory of extremes. Its application to multivariate nonstationary sequences is thus rather straightforward. Finally, Chapter 11 contains the statistical analysis of nonstationary ecological time series.
This book comes with the statistical software system XTREMES, version 1.2, produced by Sylvia Haßmann, Rolf-Dieter Reiss and Michael Thomas. We refer to the appendix (co-authored by Sylvia Haßmann and Michael Thomas) for a user’s guide to XTREMES.

MSC:
60G70 Extreme value theory; extremal stochastic processes
60-02 Research exposition (monographs, survey articles) pertaining to probability theory
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