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© 1991

Probability in Banach Spaces

Isoperimetry and Processes

  • A very comprehensive book which develops a wide variety of the methods existing in this field

  • An event for mathematicians working or interested in probability in Banach spaces

  • a presentation of the main aspects of the theory of probability in Banach spaces

Book

Part of the Ergebnisse der Mathematik und ihrer Grenzgebiete book series (volume 23)

Table of contents

  1. Front Matter
    Pages I-XII
  2. Introduction

    1. Michel Ledoux, Michel Talagrand
      Pages 1-6
  3. Notation

    1. Michel Ledoux, Michel Talagrand
      Pages 7-12
  4. Isoperimetric Background and Generalities

    1. Front Matter
      Pages 13-13
  5. Banach Space Valued Random Variables and Their Strong Limiting Properties

    1. Front Matter
      Pages 53-53
    2. Michel Ledoux, Michel Talagrand
      Pages 54-88
    3. Michel Ledoux, Michel Talagrand
      Pages 89-121
    4. Michel Ledoux, Michel Talagrand
      Pages 122-148
    5. Michel Ledoux, Michel Talagrand
      Pages 149-177
    6. Michel Ledoux, Michel Talagrand
      Pages 178-195
    7. Michel Ledoux, Michel Talagrand
      Pages 196-234
  6. Tightness of Vector Valued Random Variables and Regularity of Random Processes

    1. Front Matter
      Pages 235-235
    2. Michel Ledoux, Michel Talagrand
      Pages 236-271
    3. Michel Ledoux, Michel Talagrand
      Pages 272-296
    4. Michel Ledoux, Michel Talagrand
      Pages 297-331
    5. Michel Ledoux, Michel Talagrand
      Pages 332-364
    6. Michel Ledoux, Michel Talagrand
      Pages 365-393
    7. Michel Ledoux, Michel Talagrand
      Pages 394-420

About this book

Introduction

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.

Keywords

Banach Space Fourier series Law of large numbers Random variable differential equation entropy law of the iterated logarithm logarithm measure

Authors and affiliations

  1. 1.Institut de Recherche Mathématique Avancée, Département de MathématiqueUniversité Louis PasteurStrasbourgFrance
  2. 2.Equipe d’AnalyseUniversité de Paris VIParisFrance
  3. 3.Department of MathematicsThe Ohio State UniversityColumbusUSA

About the authors

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 stochastic 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.

Bibliographic information

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Reviews

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