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Upper and Lower Bounds for Stochastic Processes

Decomposition Theorems

  • Book
  • © 2021

Overview

Authors:
  1. Michel Talagrand

    1. Paris, France

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  • Presents complete proofs of deep results in probability
  • Provides essential tools and methods
  • Discusses a number of classical problems and fundamental open questions

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Table of contents (19 chapters)

  1. Front Matter

    Pages i-xviii
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  2. What Is This Book About?

    • Michel Talagrand
    Pages 1-17

  3. The Generic Chaining

    1. Front Matter

      Pages 19-19
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    2. Gaussian Processes and the Generic Chaining

      • Michel Talagrand
      Pages 21-86

    3. Trees and Other Measures of Size

      • Michel Talagrand
      Pages 87-109

    4. Matching Theorems

      • Michel Talagrand
      Pages 111-162

  4. Some Dreams Come True

    1. Front Matter

      Pages 163-163
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    2. Warming Up with p-Stable Processes

      • Michel Talagrand
      Pages 165-173

    3. Bernoulli Processes

      • Michel Talagrand
      Pages 175-200

    4. Random Fourier Series and Trigonometric Sums

      • Michel Talagrand
      Pages 201-286

    5. Partitioning Scheme and Families of Distances

      • Michel Talagrand
      Pages 287-308

    6. Peaky Part of Functions

      • Michel Talagrand
      Pages 309-323

    7. Proof of the Bernoulli Conjecture

      • Michel Talagrand
      Pages 325-372

    8. Random Series of Functions

      • Michel Talagrand
      Pages 373-397

    9. Infinitely Divisible Processes

      • Michel Talagrand
      Pages 399-415

    10. Unfulfilled Dreams

      • Michel Talagrand
      Pages 417-429

  5. Practicing

    1. Front Matter

      Pages 431-431
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    2. Empirical Processes, II

      • Michel Talagrand
      Pages 433-456

    3. Gaussian Chaos

      • Michel Talagrand
      Pages 457-492

    4. Convergence of Orthogonal Series: Majorizing Measures

      • Michel Talagrand
      Pages 493-547
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Keywords

About this book

This book provides an in-depth account of modern methods used to bound the supremum of stochastic processes. Starting from first principles, it takes the reader to the frontier of current research. This second edition has been completely rewritten, offering substantial improvements to the exposition and simplified proofs, as well as new results.
The book starts with a thorough account of the generic chaining, a remarkably simple and powerful method to bound a stochastic process that should belong to every probabilist’s toolkit. The effectiveness of the scheme is demonstrated by the characterization of sample boundedness of Gaussian processes. Much of the book is devoted to exploring the wealth of ideas and results generated by thirty years of efforts to extend this result to more general classes of processes, culminating in the recent solution of several key conjectures.


A large part of this unique book is devoted to the author’s influential work.While many of the results presented are rather advanced, others bear on the very foundations of probability theory. In addition to providing an invaluable reference for researchers, the book should therefore also be of interest to a wide range of readers.

Reviews

“The book includes a rich collection of exercises that will allow the reader to gain skills for a better understanding. The book is then suitable as a textbook for an advanced course. … The systematic and deep treatment of the subject under study makes the book a good reference for the specialist.” (Erick Treviño-Aguilar, Mathematical Reviews, March, 2023)

Authors and Affiliations

  • Paris, France

    Michel Talagrand

About the author

Michel Talagrand has made profound contributions to mathematics, notably in probability theory and related topics. The author of several books and well over 200 research papers, he is the recipient of several awards, including the Loève Prize, the Fermat Prize and the Shaw Prize. He has been a plenary speaker at the International Congress of Mathematicians and is a member of the French Academy of Sciences.

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