Probability in Banach Spaces: Isoperimetry and Processes

Introduction......Page 1
Chapter 1. Isoperimetric inequalities and the concentration of measure phenomenon......Page 5
1.1. Some isoperimetric inequalities on the sphere, in Gauss space and on the cube......Page 6
1.2. An isoperimetric inequality for product measures......Page 17
1.3. Martingale inequalities......Page 23
Notes and references......Page 27
2.1 Banach space valued Radon random variables......Page 31
2.2 Random processes and vector valued random variables......Page 38
2.3 Symmetric random variables and Lévy's inequalities......Page 42
2.4 Some inequalities for real random variables......Page 46
Notes and references......Page 47
Chapter 3. Gaussian random variables......Page 50
3.1. Integrability and tail behavior......Page 52
3.2. Integrability of Gaussian chaos......Page 61
3.3. Comparison theorems......Page 71
Notes and references......Page 87
4.1. Real Rademacher averages......Page 91
4.2. The contraction principle......Page 98
4.3. Integrability and tail behavior of Rademacher series......Page 101
4.4. Integrability of Rademacher chaos......Page 109
4.5. Comparison theorems......Page 116
Notes and references......Page 126
Chapter 5. Stable random variables......Page 129
5.1. Representation of stable random variables......Page 131
5.2. Integrability and tail behavior......Page 141
5.3. Comparison theorems......Page 150
Notes and references......Page 156
Chapter 6. Sums of independent random variables......Page 159
6.1 Symmetrization and some inequalities on sums of independent random variables......Page 160
6.2 Integrability of sums of independent random variables......Page 165
6.3 Concentration and tail behavior......Page 172
Notes and references......Page 187
Chapter 7. The strong law of large numbers......Page 190
7.1 A general statement for strong limit theorems......Page 191
7.2 Examples of laws of large numbers......Page 198
Notes and references......Page 207
8.1 The law of the iterated logarithm of Kolmogorov......Page 210
8.2 The law of the iterated logarithm of Hartman-Wintner-Strassen......Page 217
8.3 On the identification of the limits......Page 232
Notes and references......Page 249
9.1 l^n_p-subspaces of Banach spaces......Page 253
9.2 Type and cotype......Page 263
9.3 Some probabilistic statements in presence of type and cotype......Page 272
Notes and references......Page 288
10.1 Some general facts about the central limit theorem......Page 292
10.2 Some central limit theorems in certain Banach spaces......Page 301
10.3 A small ball criterion for the central limit theorem......Page 310
Notes and references......Page 318
Chapter 11. Regularity of random processes......Page 321
11.1 Regularity of random processes under metric entropy conditions......Page 323
11.2 Regularity of random processes under majorizing measure conditions......Page 333
11.3 Examples of applications......Page 344
Notes and references......Page 356
12.1 Regularity of Gaussian processes......Page 360
12.2 Necessary conditions for boundedness and continuity of stable processes......Page 379
12.3 Applications and conjectures on Rademacher processes......Page 388
Notes and references......Page 395
13.1. Stationarity and entropy......Page 398
13.2. Random Fourier series......Page 403
13.3. Stable random Fourier series and strongly stationary processes......Page 417
13.4. Vector valued random Fourier series......Page 423
Notes and references......Page 428
14.1. The central limit theorem for Lipschitz processes......Page 431
14.2. Empirical processes and random geometry......Page 440
14.3. Vapnik-Chervonenkis classes of sets......Page 449
Notes and references......Page 458
15.1. Subspaces of small codimension......Page 462
15.2. Conjectures on Sudakov's minoration for chaos......Page 469
15.3. An inequality of J. Bourgain......Page 472
15.4. Invertibility of submatrices......Page 476
15.5. Embedding subspaces of L_p into l^N_p......Page 480
15.6. Majorizing measures on ellipsoids......Page 492
15.7. Cotype of the canonical injection l^N_infty -> L_{2,1}......Page 497
15.8. Miscellaneous problems......Page 501
Notes and references......Page 503