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E-grāmata: Geometry and Martingales in Banach Spaces

(Case Western Reserve University, Cleveland, Ohio USA)
  • Formāts: 330 pages
  • Izdošanas datums: 12-Oct-2018
  • Izdevniecība: CRC Press
  • Valoda: eng
  • ISBN-13: 9780429868832
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Geometry and Martingales in Banach SpacesPalielināt
  • Formāts: 330 pages
  • Izdošanas datums: 12-Oct-2018
  • Izdevniecība: CRC Press
  • Valoda: eng
  • ISBN-13: 9780429868832
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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.

Recenzijas

"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 "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

Introduction ix
1 Preliminaries: Probability and geometry in Banach spaces
1(22)
1.1 Random vectors in Banach spaces
1(2)
1.2 Random series in Banach spaces
3(5)
1.3 Basic geometry of Banach spaces
8(5)
1.4 Spaces with invariant under spreading norms which are finitely representable in a given space
13(3)
1.5 Absolutely summing operators and factorization results
16(7)
2 Dentability, Radon-Nikodym Theorem, and Martingale Convergence Theorem
23(24)
2.1 Dentability
23(6)
2.2 Dentability versus Radon-Nikodym property, and martingale convergence
29(9)
2.3 Dentability and submartingales in Banach lattices and lattice bounded operators
38(9)
3 Uniform Convexity and Uniform Smoothness
47(20)
3.1 Basic concepts
47(4)
3.2 Martingales in uniformly smooth and uniformly convex spaces
51(10)
3.3 General concept of super-property
61(2)
3.4 Martingales in super-reflexive Banach spaces
63(4)
4 Spaces that do not contain c0
67(8)
4.1 Boundedness and convergence of random series
67(5)
4.2 Pre-Gaussian random vectors
72(3)
5 Cotypes of Banach spaces
75(40)
5.1 Infracotypes of Banach spaces
75(4)
5.2 Spaces of Rademacher cotype
79(6)
5.3 Local structure of spaces of cotype q
85(7)
5.4 Operators in spaces of cotype q
92(7)
5.5 Random series and law of large numbers
99(11)
5.6 Central limit theorem, law of the iterated logarithm, and infinitely divisible distributions
110(5)
6 Spaces of Rademacher and stable types
115(82)
6.1 Infratypes of Banach spaces
115(4)
6.2 Banach spaces of Rademacher-type p
119(12)
6.3 Local structures of spaces of Rademacher-type p
131(9)
6.4 Operators on Banach spaces of Rademacher-type p
140(4)
6.5 Banach spaces of stable-type p and their local structures
144(9)
6.6 Operators on spaces of stable-type p
153(6)
6.7 Extented basic inequalities and series of random vectors in spaces of type p
159(10)
6.8 Strong laws of large numbers and asymptotic behavior of random sums in spaces of Rademacher-type p
169(9)
6.9 Weak and strong laws of large numbers in spaces of stable-type p
178(4)
6.10 Random integrals, convergence of infinitely divisible measures and the central limit theorem
182(15)
7 Spaces of type 2
197(30)
7.1 Additional properties of spaces of type 2
197(5)
7.2 Gaussian random vectors
202(4)
7.3 Kolmogorov's inequality and three-series theorem
206(2)
7.4 Central limit theorem
208(10)
7.5 Law of iterated logarithm
218(5)
7.6 Spaces of type 2 and cotype 2
223(4)
8 Beck convexity
227(46)
8.1 General definitions and properties and their relationship to types of Banach spaces
227(9)
8.2 Local structure of B-convex spaces and preservation of B-convexity under standard operations
236(6)
8.3 Banach lattices and reflexivity of B-convex spaces
242(7)
8.4 Classical weak and strong laws of large numbers in B-convex spaces
249(9)
8.5 Laws of large numbers for weighted sums and not necessarily independent summands
258(5)
8.6 Ergodic properties of B-convex spaces
263(8)
8.7 Trees in B-convex spaces
271(2)
9 Marcinkiewicz-Zygmund Theorem in Banach spaces
273(24)
9.1 Preliminaries
273(3)
9.2 Brunk-Prokhorov's type strong law and related rates of convergence
276(3)
9.3 Marcinkiewicz-Zygmund type strong law and related rates of convergence
279(9)
9.4 Brunk and Marcinkiewicz-Zygmund type strong laws for martingales
288(9)
Bibliography 297(16)
Index 313
Wojbor A. Woyczyski 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.
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