Geometry and Martingales in Banach Spaces

Geometry and Martingales in Banach Spaces provides a compact exposition of the results explaining the interrelations existing between the metric geometry of Banach spaces and the theory of martingales, and general random vectors with values in those Banach spaces. Geometric concepts such as dentability, uniform smoothness, uniform convexity, Beck convexity, etc. turn out to characterize asymptotic behavior of martingales with values in Banach spaces.

Table of Contents:
Introduction 1 Preliminaries: Probability and geometry in Banach spaces 1.1 Random vectors in Banach spaces 1.2 Random series in Banach spaces 1.3 Basic geometry of Banach spaces 1.4 Spaces with invariant under spreading norms which are finitely representable in a given space 1.5 Absolutely summing operators and factorization results 2 Dentability, Radon-Nikodym Theorem, and Martingale Convergence Theorem 2.1 Dentability 2.2 Dentability vs. Radon-Nikodym Property, and Martingale Convergence 2.3 Dentability and submartingales in Banach lattices, lattice bounded operators 3 Uniform Convexity and Uniform Smoothness 3.1 Basic concepts 3.2 Martingales in uniformly smooth and uniformly convex spaces 3.3 The general concept of super-property 3.4 Martingales in super-reflexive Banach spaces 4 Spaces that do not contain c0 4.1 Boundedness and convergence of random series 4.2 The case of pre-Gaussian random vectors 5 Cotypes of Banach spaces 5.1 Infracotypes of Banach spaces 5.2 Spaces of Rademacher cotype 5.3 Local structure of spaces of cotype q 5.4 Operators in spaces of cotype q 5.5 Random series and the law of large numbers 5.6 Central Limit Theorem, Law of the Iterated Logarithm, and infinitely divisible distributions 6 Spaces of Rademacher and stable type 6.1 Infratypes of Banach spaces 6.2 Banach spaces of Rademacher-type p 6.3 Local structure of spaces of Rademacher-type p 6.4 Operators on Banach spaces of Rademacher-type p 6.5 Banach spaces of stable-type p and their local structure 6.6 Operators on spaces of stable-type p 6.7 Extented basic inequalities, and series of random vectors in spaces of type p 6.8 Strong laws of large numbers and asymptotic behavior of random sums in spaces of Rademacher-type p 6.9 Weak and strong laws of large numbers in spaces of stable-type p 6.10 Random integrals, convergence of infinitely divisible measures and the central limit theorem 7 Spaces of type 2 7.1 Additional properties of spaces of type 2 7.2 Gaussian random vectors 7.3 Kolmogorov’s inequality and the three-series theorem 7.4 Central limit theorem 7.5 Law of the iterated logarithm 7.6 Spaces of both, type 2 and cotype 2 8 Beck convexity 8.1 General definitions and properties, relationship to types of Banach spaces 8.2 Local structure of B-convex spaces and preservation of Bconvexity under standard operations 8.3 Banach lattices and reflexivity of B-convex spaces 8.4 Classical weak and strong laws of large numbers in B-convex spaces 8.5 Laws of large numbers for weighted sums and not necessarily independent summands 8.6 Ergodic properties of B-convex spaces 8.7 Trees in B-convex spaces 9 Marcinkiewicz-Zygmund Theorem in Banach spaces 9.1 Preliminaries 9.2 Brunk-Prokhorov’s type strong law and related rates of convergence 9.3 Marcinkiewicz-Zygmund type strong law and related rates of convergence 9.4 Brunk and Marcinkiewicz-Zygmund type strong laws for martingales Bibliography Index