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Introduction 1

Notation 7

Part O.Isoperimetric Background and Generalities 14

Chapter 1.Isoperimetric Inequalities and the Concentration of Measure Phenomenon 14

1.1 Some Isoperimetric Inequalities on the Sphere,in Gauss Space and on the Cube 15

1.2 An Isoperimetric Inequality for Product Measures 25

1.3 Martingale Inequalities 30

Notes and References 34

Chapter 2.Generalities on Banach Space Valued Random Variables and Random Processes 37

2.1 Banach Space Valued Radon Random Variables 37

2.2 Random Processes and Vector Valued Random Variables 43

2.3 Symmetric Random Variables and Lévy's Inequalities 47

2.4 Some Inequalities for Real Valued Random Variables 50

Notes and References 52

Part Ⅰ.Banach Space Valued Random Variables and Their Strong Limiting Properties 54

Chapter 3.Gaussian Random Variables 54

3.1 Integrability and Tail Behavior 56

3.2 Integrability of Gaussian Chaos 64

3.3 Comparison Theorems 73

Notes and References 87

Chapter 4.Rademacher Averages 89

4.1 Real Rademacher Averages 89

4.2 The Contraction Principle 95

4.3 Integrability and Tail Behavior of Rademacher Series 98

4.4 Integrability of Rademacher Chaos 104

4.5 Comparison Theorems 111

Notes and References 120

Chapter 5.Stable Random Variables 122

5.1 Representation of Stable Random Variables 124

5.2 Integrability and Tail Behavior 133

5.3 Comparison Theorems 141

Notes and References 147

Chapter 6.Sums of Independent Random Variables 149

6.1 Symmetrization and Some Inequalities for Sums of Independent Random Variables 150

6.2 Integrability of Sums of Independent Random Variables 155

6.3 Concentration and Tail Behavior 162

Notes and References 176

Chapter 7.The Strong Law of Large Numbers 178

7.1 A General Statement for Strong Limit Theorems 179

7.2 Examples of Laws of Large Numbers 186

Notes and References 195

Chapter 8.The Law of the Iterated Logarithm 196

8.1 Kolmogorov's Law of the Iterated Logarithm 196

8.2 Hartman-Wintner-Strassen's Law of the Iterated Logarithm 203

8.3 On the Identification of the Limits 216

Notes and References 233

Part Ⅱ.Tightness of Vector Valued Random Variables and Regularity of Random Processes 236

Chapter 9.Type and Cotype of Banach Spaces 236

9.1 enp-Subspaces of Banach Spaces 237

9.2 Type and Cotype 245

9.3 Some Probabilistic Statements in Presence of Type and Cotype 254

Notes and References 269

Chapter 10.The Central Limit Theorem 272

10.1 Some General Facts About the Central Limit Theorem 273

10.2 Some Central Limit Theorems in Certain Banach Spaces 280

10.3 A Small Ball Criterion for the Central Limit Theorem 289

Notes and References 295

Chapter 11.Regularity of Random Processes 297

11.1 Regularity of Random Processes Under Metric Entropy Conditions 299

11.2 Regularity of Random Processes Under Majorizing Measure Conditions 309

11.3 Examples of Applications 318

Notes and References 329

Chapter 12.Regularity of Gaussian and Stable Processes 332

12.1 Regularity of Gaussian Processes 333

12.2 Necessary Conditions for the Boundedness and Continuity of Stable Processes 349

12.3 Applications and Conjectures on Rademacher Processes 357

Notes and References 363

Chapter 13.Stationary Processes and Random Fourier Series 365

13.1 Stationarity and Entropy 365

13.2 Random Fourier Series 369

13.3 Stable Random Fourier Series and Strongly Stationary Processes 382

13.4 Vector Valued Random Fourier Series 387

Notes and References 392

Chapter 14.Empirical Process Methods in Probability in Banach Spaces 394

14.1 The Central Limit Theorem for Lipschitz Processes 395

14.2 Empirical Processes and Random Geometry 402

14.3 Vapnik-Chervonenkis Classes of Sets 411

Notes and References 419

Chapter 15.Applications to Banach Space Theory 421

15.1 Subspaces of Small Codimension 421

15.2 Conjectures on Sudakov's Minoration for Chaos 427

15.3 An Inequality of J.Bourgain 430

15.4 Invertibility of Submatrices 434

15.5 Embedding Subspaces of Lp into eNP 438

15.6 Majorizing Measures on Ellipsoids 448

15.7 Cotype of the Canonical Injectione eN∞→L2,1 453

15.8 Miscellaneous Problems 456

Notes and References 459

References 461

Subject Index 478