The fundamental question of characterizing continuity and boundedness of Gaussian processes goes back to Kolmogorov. After essential contributions by R. Dudley and X. Fernique, it was solved by the author in 1985. This advance was followed by a great improvement of our understanding of the boundedness of other fundamental classes of processes (empirical processes, infinitely divisible processes, etc.) This challenging body of work has now been considerably simplified through the notion of "generic chaining", a completely natural variation on the ideas of Kolmogorov. The entirely new presentation adopted here takes the reader from the first principles to the edge of current knowledge, and to the wonderful open problems that remain in this domain.
Overview and Basic Facts.- Gaussian Processes and Related Structures.- Matching Theorems.- The Bernoulli Conjecture.- Families of distances.- Applications to Banach Space Theory.
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)
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Inhaltsverzeichnis
Overview and Basic Facts.- Gaussian Processes and Related Structures.- Matching Theorems.- The Bernoulli Conjecture.- Families of distances.- Applications to Banach Space Theory.
Klappentext
The fundamental question of characterizing continuity and boundedness of Gaussian processes goes back to Kolmogorov. After contributions by R. Dudley and X. Fernique, it was solved by the author. This book provides an overview of "generic chaining", a completely natural variation on the ideas of Kolmogorov. It takes the reader from the first principles to the edge of current knowledge and to the open problems that remain in this domain.
