Conditional Measures and Applications

ISBN-13: 9781574445930 / Angielski / Twarda / 2005 / 506 str.

M. M. Rao; Rao Rao

Opis

The second edition of this authoritative text offers an in-depth treatment of all aspects of conditional expectations and probability measures and their structural analysis. However, what makes this the definitive reference on conditional measures is the author's keen ability to link theory with practical applications. The author covers applications in sufficiency, Markov processes and martingales. He also illustrates the difficulties in calculating conditional expectations for continuous multivariate distributions and methods for correct solutions in such cases. Each chapter concludes with several open-ended problems that reinforce principles and promote practical problem solving.

Spis treści

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

O autorze

M.M. Rao