Stopping Times and Directed Processes (Encyclopedia of Mathematics and its Applications)

Cover......Page 1
Title......Page 6
Copyright......Page 7
Contents......Page 8
Preface......Page 12
1. Stopping times......Page 16
1.1. Definitions......Page 17
1.2. The amatt convergence theorem......Page 24
1.3. Directed processes and the Radon-Nikodym theorem......Page 28
1.4. Conditional expectations......Page 34
2. Infinite measure and Orlicz spaces......Page 48
2.1. Orlicz spaces......Page 49
2.2. More on Orlicz spaces......Page 66
2.3. Uniform integrability and conditional expectation......Page 83
3.1. The three-function inequality......Page 97
3.2. Sharp maximal inequality for martingale transforms......Page 114
3.3. Prophet compared to gambler......Page 118
4. Directed index set......Page 128
4.1. Essential and stochastic convergence......Page 129
4.2. The covering condition (V)......Page 142
4.3. Lip-bounded martingales......Page 159
4.4. Ll-bounded martingales......Page 174
5. Banach-valued random variables......Page 186
5.1. Vector measures and integrals......Page 187
5.2. Martingales and amarts......Page 198
5.3. The Radon-Nikodym property......Page 213
5.4. Geometric properties......Page 233
5.5. Operator ideals......Page 247
6.1. Maximal inequalities for supermartingales......Page 269
6.2. Decompositions of submartingales......Page 282
6.3. The norm of the square function of a martingale......Page 288
6.4. Lifting......Page 295
7.1. Derivation in R......Page 306
7.2. Derivation in Rd......Page 315
7.3. Abstract derivation......Page 323
7.4. D-bases......Page 342
8. Pointwise ergodic theorems......Page 359
8.1. Preliminaries......Page 360
8.2. Weak maximal inequalities......Page 365
8.3. Hopf's decomposition......Page 370
8.4. The a-algebra of absorbing sets......Page 371
8.5. The Chacon-Ornstein theorem (conservative case)......Page 376
8.6. Superadditive processes......Page 380
9. Multiparameter processes......Page 397
9.1. A multiparameter convergence principle......Page 398
9.2. Multiparameter Cesaro averages of operators......Page 405
9.3. Multiparameter ratio ergodic theorems......Page 409
9.4. Multiparameter martingales......Page 411
References......Page 422
Index of names......Page 433
Index of terms......Page 436