Laws of Small Numbers: Extremes and Rare Events (Oberwolfach Seminars)

Since the publication of the first edition of this seminar book in 1994, the theory and applications of extremes and rare events have enjoyed an enormous and still increasing interest. The intention of the book is to give a mathematically oriented development of the theory of rare events underlying various applications. This characteristic of the book was strengthened in the second edition by incorporating various new results on about 130 additional pages. Part II, which has been added in the second edition, discusses recent developments in multivariate extreme value theory. Particularly notable is a new spectral decomposition of multivariate distributions in univariate ones which makes multivariate questions more accessible in theory and practice. One of the most innovative and fruitful topics during the last decades was the introduction of generalized Pareto distributions in the univariate extreme value theory. Such a statistical modelling of extremes is now systematically developed in the multivariate framework.

Since the publication of the first edition of this seminar book in 1994, the theory and applications of extremes and rare events have enjoyed an enormous and still increasing interest. The intention of the book is to give a mathematically oriented development of the theory of rare events underlying various applications. This characteristic of the book was strengthened in the second edition by incorporating various new results. In this third edition, the dramatic change of focus of extreme value theory has been taken into account: from concentrating on maxima of observations it has shifted to large observations, defined as exceedances over high thresholds. One emphasis of the present third edition lies on multivariate generalized Pareto distributions, their representations, properties such as their peaks-over-threshold stability, simulation, testing and estimation.

Reviews of the 2nd edition:

In brief, it is clear that this will surely be a valuable resource for anyone involved in, or seeking to master, the more mathematical features of this field.
David Stirzaker, Bulletin of the London Mathematical Society

Laws of Small Numbers can be highly recommended to everyone who is looking for a smooth introduction to Poisson approximations in EVT and other fields of probability theory and statistics. In particular, it offers an interesting view on multivariate EVT and on EVT for non-iid observations, which is not presented in a similar way in any other textbook.
Holger Drees, Metrika

Front Matter....Pages i-xiii Front Matter....Pages 1-1 Functional Laws of Small Numbers....Pages 3-22 Extreme Value Theory....Pages 23-74 Estimation of Conditional Curves....Pages 75-103 Front Matter....Pages 105-105 Basic Theory of Multivariate Maxima....Pages 107-130 Multivariate Extremes: The Pickands Approach....Pages 131-159 The Pickands Approach in the Bivariate Case....Pages 161-202 Multivariate Extremes: Supplementary Concepts and Results....Pages 203-232 Front Matter....Pages 233-233 Introduction to the Non IID Case....Pages 235-248 Extremes of Random Sequences....Pages 249-271 Extremes of Gaussian Processes....Pages 273-296 Extensions for Rare Events....Pages 297-327 Statistics of Extremes....Pages 329-345 Back Matter....Pages 347-378 "Since the publication of the first edition of this seminar book in 1994, the theory and applications of extremes and rare events have enjoyed an enormous and still increasing interest. The intention of the book is to give a mathematically oriented development of the theory of rare events underlying various applications. This characteristic of the book was strengthened in the second edition by incorporating various new results on about 130 additional pages." "This book is accessible to graduate students and researchers with basic knowledge in probability theory and, partly, in point processes and Gaussian processes. The required statistical prerequisites are minimal."--Jacket