巴拿赫空间中的概率论

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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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.