The Generic Chaining: Upper and Lower Bounds of Stochastic Processes / Edition 1

by Michel Talagrand

Overview

Author's Note:

The material of this book has been reworked and expanded with a lot more detail and published in the author's 2014 book "Upper and Lower Bounds for Shastic Processes" (Ergebnisse Vol. 60, ISBN 978-3-642-54074-5). That book is much easier to read and covers everything that is in "The Generic Chaining" book in a more detailed and comprehensible way.

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What is the maximum level a certain river is likely to reach over the next 25 years? (Having experienced three times a few feet of water in my house, I feel a keen personal interest in this question. ) There are many questions of the same nature: what is the likely magnitude of the strongest earthquake to occur during the life of a planned building, or the speed of the strongest wind a suspension bridge will have to stand? All these situations can be modeled in the same manner. The value X of the quantity of interest (be it water t level or speed of wind) at time t is a random variable. What can be said about the maximum value of X over a certain range of t? t A collection of random variables (X ), where t belongs to a certain index t set T, is called a shastic process, and the topic of this book is the study of the supremum of certain shastic processes, and more precisely to find upper and lower bounds for the quantity EsupX . (0. 1) t t?T Since T might be uncountable, some care has to be taken to define this quantity. For any reasonable definition of Esup X we have t t?T EsupX =sup{EsupX ; F?T,Ffinite} , (0. 2) t t t?T t?F an equality that we will take as the definition of the quantity Esup X . t t?T Thus, the crucial case for the estimation of the quantity (0.

Product Details

ISBN-13:	9783642063862
Publisher:	Springer Berlin Heidelberg
Publication date:	12/01/2010
Series:	Springer Monographs in Mathematics
Edition description:	2005
Pages:	222
Product dimensions:	6.10(w) x 9.25(h) x 0.02(d)

Table of Contents

Overview and Basic Facts.- Gaussian Processes and Related Structures.- Matching Theorems.- The Bernoulli Conjecture.- Families of distances.- Applications to Banach Space Theory.

Editorial Reviews

From the reviews:

"This textbook gives a systematic approach to the problem of deriving good bounds for shastic processes using the ‘generic chaining’ method … . The presentation in this book … takes the reader from a simple and intuitive explanation of the basic idea underlying the chaining technique to the edge of today’s knowledge. … The entertaining and sometimes humorous style makes this book a pleasure to read." (Dierk Peithmann, Zentralblatt MATH, Vol. 1075, 2006)

"The results discussed in the book involve a tremendous amount of creativity … . I believe that the monograph is of interest to anyone who works even remotely with indexed families of maps… . is of interest to almost everyone who works in probability, analysis, or ergodic theory even if she/he has not done any research involving the topics covered by the volume under review. … I want to emphasize the fact that the book is of interest to a larger audience … ." (Radu Zaharopol, SIAM Review, Vol. 49 (2), 2007)

From the Publisher