UID:
almafu_9959240272802883
Format:
1 online resource (xii, 428 pages) :
,
digital, PDF file(s).
ISBN:
1-139-88626-6
,
0-511-94555-8
,
1-107-10260-X
,
0-521-13508-7
,
1-107-08791-0
,
1-107-09413-5
,
0-511-57474-6
,
1-107-09101-2
Series Statement:
Encyclopedia of mathematics and its applications ; v. 47
Content:
The notion of 'stopping times' is a useful one in probability theory; it can be applied to both classical problems and fresh ones. This book presents this technique in the context of the directed set, stochastic processes indexed by directed sets, and many applications in probability, analysis and ergodic theory. Martingales and related processes are considered from several points of view. The book opens with a discussion of pointwise and stochastic convergence of processes, with concise proofs arising from the method of stochastic convergence. Later, the rewording of Vitali covering conditions in terms of stopping times clarifies connections with the theory of stochastic processes. Solutions are presented here for nearly all the open problems in the Krickeberg convergence theory for martingales and submartingales indexed by directed set. Another theme of the book is the unification of martingale and ergodic theorems.
Note:
Title from publisher's bibliographic system (viewed on 05 Oct 2015).
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Cover; Half-title; Title; Copyright; Contents; Preface; 1. Stopping times; 1.1. Definitions; Directed sets; Stochastic basis; Stopping times; Optional stopping; Complements; 1.2. The amart convergence theorem; The lattice property; Convergence; Complements; 1.3. Directed processes and the Radon-Nikodym theorem; Processes indexed by directed sets; Complements; 1.4. Conditional expectations; Definition and basic properties; Martingales and related processes; Riesz decomposition; The sequential case; Complements; 2. Infinite measure and Orlicz spaces; 2.1. Orlicz spaces
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Orlicz functions and their conjugatesOrlicz spaces; Complements; 2.2. More on Orlicz spaces; Comparison of orlicz spaces; Largest and smallest orlicz functions; Duality for orlicz spaces; 2.3. Uniform integrability and conditional expectation; Conditional expectation in infinite measure spaces; Complements; 3. Inequalities; 3.1. The three-function inequality; Complements; 3.2. Sharp maximal inequality for martingale transforms; 3.3. Prophet compared to gambler; Stopped processes; Transformed processes; The case of signed U; Complements; Remarks; 4. Directed index set
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4.1. Essential and stochastic convergenceEssential convergence; Snell envelope; Complements; 4.2. The covering condition (V); Condition (V); Example: Totally ordered basis; Example: Interval partitions; Essential convergence; Complements; 4.3. Lψ-bounded martingales; Multivalued stopping times; Covering condition (VΦ); Necessity of (VΦ); Functional condition (FVΦ); 4.4. L1-bounded martingales; A counterexample; Complements; 5. Banach-valued random variables; 5.1. Vector measures and integrals; Theorems of functional analysis; Vector measures; The radon-nikodym property
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5.2. Martingales and amartsElementary properties; Complements; 5.3. The Radon-Nikodým property; Scalar and pettis norm convergence; Weak a.s. convergence; Strong convergence; T-convergence; Converses; Complements; 5.4. Geometric properties; The choquet-edgar theorem; Common fixed points for noncommuting maps; Dentability; Strongly exposed points; Complements; Remarks; 5.5. Operator ideals; Absolutely summing operators; Radon-nikodym operators; Asplund operators; Complements; 6. Martingales; 6.1. Maximal inequalities for supermartingales; A maximal inequality; A law of large numbers
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A generalization of independence: star-mixingComplements; Remarks; 6.2. Decompositions of submartingales; Doob's decomposition; Martingale transforms; Complements; 6.3. The norm of the square function of a martingale; Remarks; 6.4. Lifting; Complements; 7. Derivation; 7.1. Derivation in R; Stochastic bases; Derivation; Remarks; 7.2. Derivation in Rd; Substantial sets; Disks and cubes; Intervals; Complements; Remarks; 7.3. Abstract derivation; Nonmeasurable sets; Derivation bases; Vitali covers and derivation; The strong vitali property; The weak vitali property; Property (FVΦ); Complements
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7.4. D-bases
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English
Additional Edition:
ISBN 0-521-35023-9
Additional Edition:
ISBN 1-306-14838-3
Language:
English
Subjects:
Mathematics
URL:
https://doi.org/10.1017/CBO9780511574740
