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Publisher:

Springer

Publication Date:

2011

Number of Pages:

480

Format:

Paperback

Series:

Classics in Mathematics

Price:

74.95

ISBN:

9783642202117

Category:

Monograph

[Reviewed by , on ]

Fernando Q. Gouvêa

01/16/2012

Springer’s *Classics in Mathematics* series offers paperback reprints of older books that have become established as classics in their fields. *Probability in Banach Spaces* was first published in 1991. The Telegraphic Review in the April 1992 of the *American Mathematical Monthly* said

An attempt to summarize the explosion of developments in the past twenty years. Focuses on two related topics: isoperimetric inequalities/methods, and the regularity of random processes. Highly technical. Contains a huge bibliography. Note price. TAV

“TAV” was Thomas A. Vessey, then a professor at St. Olaf College. The price he was “noting” was indeed spectacular for the time: $129. Today, that price would be less unusual, though still high. This edition comes in at a much smaller nominal price, which is good news.

The book is reprinted essentially unchanged. A welcome addition to the front matter is a page with photos and brief biographies of the authors. Surprisingly, the final page in the original edition, which was an advertisement for other books in the third seris of Springer’s *Ergebnisse*, is included as well. There were 22 books in the series at that time, compared to 57 listed today on Springer’s web site.

The back cover quotes a review from *MathSciNet* by Evarist Giné:

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 during the last thirty years. There is no doubt in the reviewer’s mind that this book will become a classic. [MR1102015 (93c:60001)]

They make one change, however, replacing “will” in the last sentence with “has.” Well, I guess it has, but that wasn’t what the review said.

The last page in this edition is also new, giving a listing of the books in Springer’s *Classics* series. They have been chosen well: as far as I can tell, the books listed are indeed worthy of inclusion in a series with that title. So is this one.

Fernando Q. Gouvêa loves books, which is a good thing, since he is the editor of MAA Reviews. In real life, he is Carter Professor of Mathematics at Colby College in Waterville, ME.

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

References

Subject Index

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