This book tries to present some of the main aspects of the
theory of Probability in Banach spaces， from the foundations of the
topic to the latest developments and current research questions.
The past twenty years saw intense activity in the study of
classical Probability Theory on infinite dimensional spaces. vector
valued random variables， boundedness and continuity of ran-dom
processes， with a fruitful interaction with classical Banach spaces
and their geometry. A large community of mathematicians， from
classical probabilists to pure analysts and functional analysts，
participated to this common achievement.
The recent use of isoperimetric tools and concentration of
measure phenomena， and of abstract random process techniques has
led today to rather a complete picture of the field. These
developments prompted the authors to undertake the writing of this
exposition based on this modern point of view.
This book does not pretend to cover all the aspects of the
subject and of its connections with other fields. In spite of its
ommissions， imperfections and errors， for which we would like to
apologize， we hope that this work gives an attractive picture of
the subject and will serve it appropriately.
目錄：
Introduction
Notation
Part 0. Isoperimetric Background and Generalities
Chapter 1. Isoperimetric Inequalities and the Concentration of
Measure Phenomenon
1.1 ome Isoperimetric Inequalities on the Sphere, in Gauss
Space and on the Cube
1.2 An Isoperimetric Inequality for Product Measures
1.3 Martingale Inequalities
Notes and References
Chapter 2. Generalities on Banach Space Valued Random Variables and
Random Processes
2.1 Banach Space Valued Radon Random Variables
2.2 Random Processes and Vector Valued R,a,ndom Variables
2.3 Symmetric Random Variables and Levy''s Inequalities
2.4 Some Inequalities for Real Valued Random Variables
Notes and References
Part I. Banach Space Valued Random Variables and Their Strong
Limiting Properties
Chapter 3. Gaussian Random Variables
3.1 Integrability and Tail Behavior
3.2 Integrability of Gaussian Chaos
3.3 Comparison Theorems
Notes and References
Chapter 4. Rademacher Averages
4.1 Real Rademacher A''verages
4.2 The Contraction Principle
4,3 Integrability and Tail Behavior of Rademacher Series
4.4 Integrability of Rademacher Chaos
4.5 Comparison Theorems
Notes and References
Chapter 5. Stable Random Variables
5.1 R;epresentation of Stable Random Variables
5.2 Integrability and Tail Behavior
5.3 Comparison Theorems
Notes and References
Chapter 6. Sums of Independent Random Variables
6.1 Symmetrization and Some Inequalities for Sums of Independent
Random Variables
6.2 Integrability of Sums of Independent Random Variables
6.3 Concentration and Tail Behavior
Notes and R,eferences
Chapter 7. The Strong Law of Large Numbers
7.1 A General Statement for Strong Limit Theorems
7.2 Examples of Laws of Large Numbers
Notes and References
Chapter 8. The Law of the lterated Logarithm
8.1 Kolmogorov''s Law of the Iterated Logarithm
8.2 Hartman-Wintner-Strassen''s Law of the Iterated Logarithm
8.3 On the Identification of the Limits
Notes and References
Part II. Tightness of Vector Valued R,andom Variables and
Regularity of Random Processes
……
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