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2nd Edition

Conditional Measures and Applications




ISBN 9781574445930
Published May 25, 2005 by Chapman and Hall/CRC
506 Pages

 
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Book Description

In response to unanswered difficulties in the generalized case of conditional expectation and to treat the topic in a well-deservedly thorough manner, M.M. Rao gave us the highly successful first edition of Conditional Measures and Applications. Until this groundbreaking work, conditional probability was relegated to scattered journal articles and mere chapters in larger works on probability. This second edition continues to offer a thorough treatment of conditioning while adding substantial new information on developments and applications that have emerged over the past decade.

Conditional Measures and Applications, Second Edition clearly elucidates the subject, from fundamental principles to abstract analysis. The author illustrates the computational difficulties in evaluating conditional probabilities in nondiscrete cases with numerous examples, demonstrates applications to Markov processes, martingales, potential theory, and Reynolds operators as well as sufficiency in statistics, and clarifies ideas in modern noncommutative probability structures through conditioning in general structures, including parts of operator algebras and "free" random variables. He also discusses existence and construction problems from the Bishop-Brouwer constructive analysis point of view.

With open problems in every chapter and links to other areas of mathematics, this invaluable second edition offers complete coverage of conditional probability and expectation and their structural analysis, from simple to advanced abstract levels, for both novices and seasoned mathematicians.

Table of Contents

THE CONCEPT OF CONDITIONING
Introduction
Conditional Probability Given a Partition
Conditional Expectation: Elementary Case
Conditioning with Densities
Conditional Probability Spaces: First Steps
Bibliographical Notes
THE KOLMOGOROV FORMULATION AND ITS PROPERTIES
Introduction of the General Concept
Basic Properties of Conditional Expectations
Conditional Probabilities in the General Case
Remarks on the Inclusion of Previous Concepts
Conditional Independence and Related Concepts
Bibliographical Notes
COMPUTATIONAL PROBLEMS ASSOCIATED WITH CONDITIONING
Introduction
Some Examples with Multiple Solutions: Paradoxes
Dissection of Paradoxes
Some Methods of Computation
Remarks on Traditional Calculations of Conditional Measures
Bibliographical Notes
AN AXIOMATIC APPROACH TO CONDITIONAL PROBABILITY
Introduction
Axiomatization of Conditioning Based on Partitions
Structure of the New Conditional Probability Functions
Some Applications
Difficulties with Earlier Examples Persist
Bibliographical Notes
REGULARITY OF CONDITIONAL MEASURES
Introduction
Existence of Regular Conditional Probabilities: Generalities
Special Spaces Admitting Regular Conditional Probabilities
Disintegration of Probability Measures and Regular Conditioning
Further Results on Disintegration
Evaluation of Conditional Expectations by Fourier Analysis
Further Evaluations of Conditional Expectations
Bibliographical Notes
SUFFICIENCY
Introduction
Conditioning Relative to Families of Measures
Sufficiency: The Dominated Case
Sufficiency: The Undominated Case
Sufficiency: Another Approach to the Undominated Case
Bibliographical Notes
ABSTRACTION OF KOLMOGOROV'S FORMULATION
Introduction
Integration Relative to Conditional Measures and Function Spaces
Functional Characterizations of Conditioning
Integral Representations of Conditional Expectations
Rényi's Formulation as a Specialization of the Abstract Version
Conditional Measures and Differentiation
Bibliographical Notes
PRODUCTS OF CONDITIONAL MEASURES
Introduction
A General Formulation of Products
General Projective Limit Theorems
Some Consequences
Remarks on Conditioning, Disintegration, and Lifting
Bibliographical Notes
APPLICATIONS TO MARTINGALES AND MARKOV PROCESSES
Introduction
Set Martingales
Martingale Convergence
Markov Processes: Some Basic Results
Further Properties of Markov Processes
Bibliographical Notes
APPLICATIONS TO MODERN ANALYSIS
Introduction and Motivation
Conditional Measures and Potential Kernels
Reynolds Operators and Conditional Expectations
Bistochastic Operators and Conditioning
Contractive Projections and Conditional Expectations
Bibliographical Notes
CONDITIONING IN GENERAL STRUCTURES
Introduction
Averagings in Cones of Positive Functions
Averaging Operators on Function Algebras
Conditioning in Operator Algebras
Free Independence and a Bijection in Operator Algebras
Some Applications of Noncommutative Conditioning
Bibliographical Notes
REFERENCES
NOTATIONS
AUTHOR INDEX
SUBJECT INDEX

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