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** 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.

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

### Biography

**Wojbor A. Woyczyński** received his PhD in Mathematics in 1968 from Wroclaw University, Poland. He moved to the U.S. in 1970, and since 1982, has been Professor of Mathematics and Statistics at Case Western Reserve University in Cleveland, where he served as chairman of the department from 1982 to 1991, and from 2001 to 2002. He has held tenured faculty positions at Wroclaw University, Poland, and at Cleveland State University, and visiting appointments at Carnegie-Mellon University, and Northwestern University. He has also given invited lecture series on short-term research visits at University of North Carolina, University of South Carolina, University of Paris, Gottingen University, Aarhus University, Nagoya University, University of Tokyo, University of Minnesota, the National University of Taiwan, Taipei, Humboldt University in Berlin, Germany, and the University of New South Wales in Sydney. He is also (co-)author and/or editor of fifteen books on probability theory, harmonic and functional analysis, and applied mathematics, and currently serves as a member of the editorial board of the *Applicationes Mathematicae*, Springer monograph series UTX, and as a managing editor of the journal *Probability and Mathematical Statistics*. His research interests include probability theory, stochastic models, functional analysis and partial differential equations and their applications in statistics, statistical physics, surface chemistry, hydrodynamics and biomedicine in which he has published about 200 research papers. He has been the advisor of more than 40 graduate students. Among other honors, in 2013 he was awarded Paris Prix la Recherche, Laureat Mathematiques, for work on mathematical evolution theory. He is currently Professor of Mathematics, Applied Mathematics and Statistics, and Director of the Case Center for Stochastic and Chaotic Processes in Science and Technology at Case Western Reserve University, in Cleveland, Ohio, U.S.A.

"The author provides detailed proofs of all the results concerning the interplay between the geometry and martingales. For purely geometric or probabilistic results only references are given, the prerequisites being familiarity with basic facts of functional analysis and probability theory. The book is of interest for researchers in Banach spaces, probability theory and their applications to the analysis of vector functions."

-Stefan Cobzas,Babes-Bolyai University, Department of Mathematics, Romania"In the 1970s there was frenetic activity in the field of probability in Banach spaces, propelled by mathematicians like, to name but a few, A. Araujo, P. Assouad, E. Gine, J. Hoffmann-Jorgensen, G. Pisier, L. Schwartz, N.N. Vakhaniya, J. Zinn and of course the present author,W. Woyczynski. The monograph under review sums up many of the results obtained in thisdecade, highlighting the interplay between probabilistic ideas and properties of Banach spaces,specifcally the Radon-Nikodym property (RNP) and local properties; indeed, the local theory of Banach spaces emerged as a result of these activities."

-Dirk Werner, Freie Universität Berlin, Berlin