2nd Edition

Conditional Measures and Applications

By M.M. Rao Copyright 2005
    506 Pages
    by Chapman & Hall

    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.

    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

    Biography

    M.M. Rao