The Generic Chaining: Upper and Lower Bounds of Stochastic Processes

ISBN-13: 9783642063862 / Angielski / Miękka / 2010 / 222 str.

Talagrand, Michel

Opis

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 ofthe strongestwind a suspension bridge will have to stand? All these situations can be modeled inthesamemanner. Thevalue 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 stochastic process, and the topic of this book is the study of the supremum of certain stochastic processes, and more precisely to ?nd 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 de?ne this quantity. For any reasonable de?nition of Esup X we have t t?T EsupX =sup{EsupX; F?T, F ?nite}, (0. 2) t t t?T t?F an equality that we will take as the de?nition of the quantity Esup X . t t?T Thus, the crucial case for the estimation of the quantity (0.

Recenzje

From the reviews:

"This textbook gives a systematic approach to the problem of deriving good bounds for stochastic 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)

Spis treści

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