Geometry and martingales in Banach spaces

Cover......Page 1
Half title......Page 2
Title......Page 4
Copyrights......Page 5
Contents......Page 6
Introduction......Page 10
Notation......Page 13
 1.1 Random vectors in Banach spaces......Page 16
 1.2 Random series in Banach spaces......Page 18
 1.3 Basic geometry of Banach spaces......Page 23
 1.4 Spaces with invariant under spreading norms which are finitely representable in a given space......Page 28
 1.5 Absolutely summing operators and factorization results......Page 31
 2.1 Dentability......Page 38
 2.2 Dentability versus Radon-Nikodym property, and martingale convergence......Page 44
 2.3 Dentability and submartingales in Banach lattices and lattice bounded operators......Page 53
 3.1 Basic concepts......Page 62
 3.2 Martingales in uniformly smooth and uniformly convex spaces......Page 66
 3.3 General concept of super-property......Page 76
 3.4 Martingales in super-reflexive Banach spaces......Page 78
 4.1 Boundedness and convergence of random series......Page 82
 4.2 Pre-Gaussian random vectors......Page 87
 5.1 Infracotypes of Banach spaces......Page 90
 5.2 Spaces of Rademacher cotype......Page 94
 5.3 Local structure of spaces of cotype q......Page 100
 5.4 Operators in spaces of cotype q......Page 107
 5.5 Random series and law of large numbers......Page 114
 5.6 Central limit theorem, law of the iterated loga-rithm, and infinitely divisible distributions......Page 125
 6.1 Infratypes of Banach spaces......Page 130
 6.2 Banach spaces of Rademacher-type p......Page 134
 6.3 Local structures of spaces of Rademacher-type p . .......Page 146
 6.4 Operators on Banach spaces of Rademacher-type p......Page 155
 6.5 Banach spaces of stable-type p and their local structures......Page 159
 6.6 Operators on spaces of stable-type p......Page 168
 6.7 Extented basic inequalities and series of random vectors in spaces of type p......Page 174
 6.8 Strong laws of large numbers and asymptotic be-havior of random sums in spaces of Rademacher-type p......Page 184
 6.9 Weak and strong laws of large numbers in spaces of stable-type p......Page 193
 6.10 Random integrals, convergence of infinitely divisi-ble measures and the central limit theorem......Page 197
 7.1 Additional properties of spaces of type 2......Page 212
 7.2 Gaussian random vectors......Page 217
 7.3 Kolmogorov’s inequality and three-series theorem .......Page 221
 7.4 Central limit theorem......Page 223
 7.5 Law of iterated logarithm......Page 233
7.6 Spaces of type 2 and cotype 2......Page 238
 8.1 General definitions and properties and their rela-tionship to types of Banach spaces......Page 242
 8.2 Local structure of B-convex spaces and preservation of B-convexity under standard operations......Page 251
 8.3 Banach lattices and reflexivity of B-convex spaces......Page 257
 8.4 Classical weak and strong laws of large numbers in B-convex spaces......Page 264
 8.5 Laws of large numbers for weighted sums and not necessarily independent summands......Page 273
 8.6 Ergodic properties of B-convex spaces......Page 278
 8.7 Trees in B-convex spaces......Page 286
 9.1 Preliminaries......Page 288
 9.2 Brunk-Prokhorov’s type strong law and related rates of convergence......Page 291
 9.3 Marcinkiewicz-Zygmund type strong law and re-lated rates of convergence......Page 294
 9.4 Brunk and Marcinkiewicz-Zygmund type strong laws for martingales......Page 303
Bibliography......Page 312
Index......Page 329