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Extreme Values, Regular Variation and Point Processes : Springer Series in Operations Research and Financial Engineering - Sidney I. Resnick

Extreme Values, Regular Variation and Point Processes

Springer Series in Operations Research and Financial Engineering

Paperback Published: 1st October 2007
ISBN: 9780387759524
Number Of Pages: 320

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Extremes Values, Regular Variation and Point Processes is a readable and efficient account of the fundamental mathematical and stochastic process techniques needed to study the behavior of extreme values of phenomena based on independent and identically distributed random variables and vectors. It presents a coherent treatment of the distributional and sample path fundamental properties of extremes and records. It emphasizes the core primacy of three topics necessary for understanding extremes: the analytical theory of regularly varying functions; the probabilistic theory of point processes and random measures; and the link to asymptotic distribution approximations provided by the theory of weak convergence of probability measures in metric spaces.

The book is self-contained and requires an introductory measure-theoretic course in probability as a prerequisite. Almost all sections have an extensive list of exercises which extend developments in the text, offer alternate approaches, test mastery and provide for enjoyable muscle flexing by a reader. The material is aimed at students and researchers in probability, statistics, financial engineering, mathematics, operations research, civil engineering and economics who need to know about: asymptotic methods for extremes; models for records and record frequencies; stochastic process and point process methods and their applications to obtaining distributional approximations; pervasive applications of the theory of regular variation in probability theory, statistics and financial engineering.

"This book is written in a very lucid way. The style is sober, the mathematics tone is pleasantly conversational, convincing and enthusiastic. A beautiful book "

Bulletin of the Dutch Mathematical Society

"This monograph is written in a very attractive style. It contains a lot of complementary exercises and practically all important bibliographical reference."

Revue Roumaine de Mathematiques Pures et Appliquees

"This book is written in a very lucid way. The style is sober, the mathematics tone is pleasantly conversational, convincing and enthusiastic. A beautiful book!"

---Bulletin of the Dutch Mathematical Society

"This monograph is written in a very attractive style. It contains a lot of complementary exercises and practically all important bibliographical reference."

---Revue Roumaine de Mathematiques Pures et Appliquees

From the reviews:

"This book provides an in-depth treatment of the theory of extreme values. ... written at a level suited for researchers and advanced graduate students in areas such as probability statistics, and operations research. ... clearly written and provides a solid and well-organised account of the theory. ... In summary, those interested in the theory will find the book most interesting. ... an excellent and clear book to read. It is a classic text ... ." (J. Shortle, Journal of the Operational Research Society, Vol. 61, 2010)

Preface to the Soft Cover Editionp. v
Preface to the Hard Cover Editionp. vii
Preliminariesp. 1
Uniform Convergencep. 1
Inverses of Monotone Functionsp. 3
Convergence to Types Theorem and Limit Distributions of Maximap. 7
Regularly Varying Functions of a Real Variablep. 12
Basicsp. 13
Deeper Results; Karamata's Theoremp. 16
Extensions of Regular Variation: [Pi]-Variation, [Gamma]-Variationp. 26
Domains of Attraction and Norming Constantsp. 38
Domain of Attraction of [Lambda](x) = exp{-e[superscript -x]}p. 38
Domain of Attraction of [Phi subscript alpha] = exp{-x[superscript -alpha]}, x > 0p. 54
Domain of Attraction of [Psi subscript alpha](x) = exp{-(-x)[superscript alpha]}, x < 0p. 59
Von Mises Conditionsp. 62
Equivalence Classes and Computation of Normalizing Constantsp. 67
Quality of Convergencep. 76
Moment Convergencep. 76
Density Convergencep. 85
Large Deviationsp. 94
Uniform Rates of Convergence to Extreme Value Lawsp. 107
Uniform Rates of Convergence to [Phi subscript alpha](x)p. 107
Uniform Rates of Convergence to [Lambda](x)p. 114
Point Processesp. 123
Fundamentalsp. 123
Laplace Functionalsp. 128
Poisson Processesp. 130
Definition and Constructionp. 130
Transformations of Poisson Processesp. 134
Vague Convergencep. 139
Weak Convergence of Point Processes and Random Measuresp. 150
Records and Extremal Processesp. 162
Structure of Recordsp. 164
Limit Laws for Recordsp. 174
Extremal Processesp. 179
Weak Convergence to Extremal Processesp. 196
Skorohod Spacesp. 196
Weak Convergence of Maximal Processes to Extremal Processes via Weak Convergence of Induced Point Processesp. 209
Extreme Value Theory for Moving Averagesp. 224
Independence of k-Record Processesp. 242
Multivariate Extremesp. 250
Max-Infinite Divisibilityp. 251
An Example: The Bivariate Normalp. 255
Characterizing Max-id Distributionsp. 257
Limit Distributions for Multivariate Extremesp. 263
Characterizing Max-Stable Distributionsp. 266
Domains of Attraction; Multivariate Regular Variationp. 276
Independence and Dependencep. 290
Associationp. 298
Referencesp. 307
Indexp. 315
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9780387759524
ISBN-10: 0387759522
Series: Springer Series in Operations Research and Financial Engineering
Audience: General
Format: Paperback
Language: English
Number Of Pages: 320
Published: 1st October 2007
Publisher: Springer-Verlag New York Inc.
Country of Publication: US
Dimensions (cm): 23.57 x 16.23  x 1.58
Weight (kg): 0.48