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Probability in Banach Spaces: Isoperimetry and Processes

Michel Ledoux and Michel Talagrand
Publication Date: 
Number of Pages: 
Classics in Mathematics
[Reviewed by
Fernando Q. Gouvêa
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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.


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

Subject Index