Probability in Banach Spaces - Isoperimetry and Processes

Based on recent developments, such as new isoperimetric inequalities and random process techniques, this book presents a thorough treatment of the main aspects of Probability in Banach spaces, and of some of their links to Geometry of Banach spaces.

Klappentext:

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. From the 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

Inhaltsverzeichnis:

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

Rezension:

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