Geometry and Martingales in Banach Spaces 1st Edition Wojbor A Woyczynski

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Product details:

• ISBN-10 ‏ : ‎ 1138616370
• ISBN-13 ‏ : ‎ 978-1138616370
• Author: Wojbor A Woyczynski

Table of contents:

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 versus Radon-Nikodym property, and martingale convergence

2.3 Dentability and submartingales in Banach lattices and 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 General concept of super-property

3.4 Martingales in super-reflexive Banach spaces

4 Spaces that do not contain

4.1 Boundedness and convergence of random series

4.2 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 law of large numbers

5.6 Central limit theorem, law of the iterated logarithm, and infinitely divisible distributions

6 Spaces of Rademacher and stable types

6.1 Infratypes of Banach spaces

6.2 Banach spaces of Rademacher-type p

6.3 Local structures 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 structures

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 three-series theorem

7.4 Central limit theorem

7.5 Law of iterated logarithm

7.6 Spaces of type 2 and cotype 2

8 Beck convexity

8.1 General definitions and properties and their relationship to types of Banach spaces

8.2 Local structure of B-convex spaces and preservation of B-convexity 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

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