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