UID:
almafu_9959239106202883
Format:
1 online resource (xvi, 351 pages) :
,
digital, PDF file(s).
ISBN:
1-139-88423-9
,
1-107-36623-2
,
1-107-37096-5
,
1-107-36132-X
,
1-107-36912-6
,
1-299-40403-0
,
1-107-36377-2
,
0-511-52617-2
Series Statement:
London Mathematical Society lecture note series ; 100
Content:
This book considers convergence of adapted sequences of real and Banach space-valued integrable functions, emphasizing the use of stopping time techniques. Not only are highly specialized results given, but also elementary applications of these results. The book starts by discussing the convergence theory of martingales and sub-( or super-) martingales with values in a Banach space with or without the Radon-Nikodym property. Several inequalities which are of use in the study of the convergence of more general adapted sequence such as (uniform) amarts, mils and pramarts are proved and sub- and superpramarts are discussed and applied to the convergence of pramarts. Most of the results have a strong relationship with (or in fact are characterizations of) topological or geometrical properties of Banach spaces. The book will interest research and graduate students in probability theory, functional analysis and measure theory, as well as proving a useful textbook for specialized courses on martingale theory.
Note:
Title from publisher's bibliographic system (viewed on 05 Oct 2015).
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Cover; Title; Copyright; Contents; Preface; Chapter I : Types of convergence; I.1. Introduction; I.1.1. Measurable functions; I.1.2. Integrable functions; I.2. Adapted sequences; I.2.1. Definition; I.2.2. Conditional expectations; I.3. Convergence; I.3.1. Pointwise convergence; I.3.2. Mean convergence; I.3.3. Pettis convergence; I.3.4. Convergence in probability; I.3.5. Convergence in probability in the stopping time sense; I.4. Notes and remarks; Chapter II : Martingale convergence theorems; II.1. Elementary results; II.2. Main results
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II.3. Convergence of martingales in general Banach spacesII.4. Notes and remarks; Chapter III : Sub- and supermartingale convergence theorems; III.1. Preliminary results; III.2. Heinich's theorem on the convergence of positive submartingales; III.3. Convergence of general submartingales; III.4. Convergence of supermartingales; III.5. Submartingale convergence in Banach lattices without (RNP); III.6. Notes and remarks; Chapter IV : Basic inequalities for adapted sequences; IV.1. Basic inequalities; IV.2. Failure of the inequalities; IV.3. Notes and remarks
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Chapter V : Convergence of generalized martingales in Banach spaces - the mean wayV.I. Uniform amarts; V.2. Amarts; V.3. Weak sequential amarts; V.4. Weak amarts; V.5. Semiamarts; V.6. Notes and remarks; Chapter VI : General directed index sets and applications of amart theory; VI.1. Convergence of adapted nets; VI.2. Applications of amart convergence results; VI.3. Notes and remarks; Chapter VII : Disadvantages of amarts. Convergence of generalized martingales in Banach spaces - the pointwise way; VII.1. Disadvantages of amarts; VII.2. Pramarts, mils, GFT; VII.3. Notes and remarks
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Chapter VIII : Convergence of generalized sub- and supermartingales in Banach latticesVIII.1. Subpramarts, superpramarts and related notions; VIII.2. Applications to pramartconvergence; VIII.3. Notes and remarks; Chapter IX : Closing remarks; IX.1. A general remark concerning scalar convergence; IX.2. Summary of the most important convergence results; IX.3. Convergence of adapted sequences of Pettis integrable functions; References; List of notations; Subject index
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English
Additional Edition:
ISBN 0-521-31715-0
Language:
English
Subjects:
Mathematics
URL:
https://doi.org/10.1017/CBO9780511526176
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