Introduction

Notation

Part O.Isoperimetric Background and Generalities

Chapter 1.Isoperimetric Inequalities and the Concentration of Measure Phenomenon

1.1 Some 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 Random Variables

2.3 Symmetric Random Variables and Lévy'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 Averages

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 Representation 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 References

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 Iterated 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 Random Variables and Regularity of Random Processes

Chapter 9.Type and Cotype of Banach Spaces

9.1 enp-Subspaces of Banach Spaces

9.2 Type and Cotype

9.3 Some Probabilistic Statements in Presence of Type and Cotype

Notes and References

Chapter 10.The Central Limit Theorem

10.1 Some General Facts About the Central Limit Theorem

10.2 Some Central Limit Theorems in Certain Banach Spaces

10.3 A Small Ball Criterion for the Central Limit Theorem

Notes and References

Chapter 11.Regularity of Random Processes

11.1 Regularity of Random Processes Under Metric Entropy Conditions

11.2 Regularity of Random Processes Under Majorizing Measure Conditions

11.3 Examples of Applications

Notes and References

Chapter 12.Regularity of Gaussian and Stable Processes

12.1 Regularity of Gaussian Processes

12.2 Necessary Conditions for the Boundedness and Continuity of Stable Processes

12.3 Applications and Conjectures on Rademacher Processes

Notes and References

Chapter 13.Stationary Processes and Random Fourier Series

13.1 Stationarity and Entropy

13.2 Random Fourier Series

13.3 Stable Random Fourier Series and Strongly Stationary Processes

13.4 Vector Valued Random Fourier Series

Notes and References

Chapter 14.Empirical Process Methods in Probability in Banach Spaces

14.1 The Central Limit Theorem for Lipschitz Processes

14.2 Empirical Processes and Random Geometry

14.3 Vapnik-Chervonenkis Classes of Sets

Notes and References

Chapter 15.Applications to Banach Space Theory

15.1 Subspaces of Small Codimension

15.2 Conjectures on Sudakov's Minoration for Chaos

15.3 An Inequality of J.Bourgain

15.4 Invertibility of Submatrices

15.5 Embedding Subspaces of Lp into eNP

15.6 Majorizing Measures on Ellipsoids

15.7 Cotype of the Canonical Injectione eN∞→L2,1

15.8 Miscellaneous Problems

Notes and References

References

Subject Index