This book examines 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 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.
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 Mathématiques Pures et Appliquées
