Probability in Banach Spaces: Isoperimetry and Processes / Edition 1

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Overview

Isoperimetric, measure concentration and random process techniques appear at the basis of the modern understanding of Probability in Banach spaces. Based on these tools, the book presents a complete treatment of the main aspects of Probability in Banach spaces (integrability and limit theorems for vector valued random variables, boundedness and continuity of random processes) and of some of their links to Geometry of Banach spaces (via the type and cotype properties). Its purpose is to present some of the main aspects of this theory, from the foundations to the most important achievements. The main features of the investigation are the systematic use of isoperimetry and concentration of measure and abstract random process techniques (entropy and majorizing measures). Examples of these probabilistic tools and ideas to classical Banach space theory are further developed.

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Editorial Reviews

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This book gives an excellent, almost complete account of the whole subject of probability in Banach spaces, a branch of probability theory that has undergone vigorous development... There is no doubt in the reviewer's mind that this book [has] become a classic.

MathSciNet

As the authors state, "this book tries to present some of the main aspects of the theory of probability in Banach spaces, from the foundation of the topic to the latest developments and current research questions''. The authors have succeeded admirably… This very comprehensive book develops a wide variety of the methods existing … in probability in Banach spaces. … It [has] become an event for mathematicians…

Zentralblatt MATH

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Product Details

Meet the Author

Michel Ledoux held first a research position with CNRS, and since 1991 is Professor at the University of Toulouse. He is moreover, since 2010, a senior member of the Institut Universitaire de France, having been also a junior member from 1997 to 2002. He has held associate editor appointments for various journals, including the Annals of Probability and Probability Theory and Related Fields (current). His research interests centre on probability, random matrices, logarithmic Sobolev inequalities, probability in Banach spaces.

Michel Talagrand has held a research position with the CNRS since 1974. His thesis was directed by Gustave Choquet and his interests revolve around the theory of shastic processes and probability in Banach spaces, as well as the mathematical theory of spin glasses. He was invited to deliver a lecture at the International Congress of Mathematicians in 1990, and to deliver a plenary lecture at the same congress in 1998. He received the Loeve Prize (1995) and the Fermat Prize (1997) for his work in probability theory. He was elected to the Paris Academy of Sciences in 2004.

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Table of Contents

Introduction.- Notation.- Part 0. Isoperimetric Background and Generalities.- Chapter 1. Isoperimetric Inequalities and the Concentration of Measure Phenomenon.- Chapter 2. Generalities on Banach Space Valued Random Variables and Random Processes.- Part I. Banach Space Valued Random Variables and Their Strong Limiting Properties.- Chapter 3. Gaussian Random Variables.- Chapter 4. Rademacher Averages.- Chapter 5. Stable Random Variables.- Chapter 6. Sums of Independent Random Variables.- Chapter 7. The Strong Law of Large Numbers.- Chapter 8. The Law of the Iterated Logarithm.- Part II. Tightness of Vector Valued Random Variables and Regularity of Random Processes.- Chapter 9. Type and Cotype of Banach Spaces.- Chapter 10. The Central Limit Theorem.- Chapter 11. Regularity of Random Processes.- Chapter 12. Regularity of Gaussian and Stable Processes.- Chapter 13. Stationary Processes and Random Fourier Series.- Chapter 14. Empirical Process Methods in Probability in Banach Spaces.- Chapter 15. Applications to Banach Space Theory.

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