Probability, random processes, and ergodic properties


Probability, random processes, and ergodic properties

Robert M. Gray

Springer, c2009

2nd ed

大学図書館所蔵 件 / 8



Includes bibliographical references and index



Probability, Random Processes, and Ergodic Properties is for mathematically inclined information/communication theorists and people working in signal processing. It will also interest those working with random or stochastic processes, including mathematicians, statisticians, and economists. Highlights: Complete tour of book and guidelines for use given in Introduction, so readers can see at a glance the topics of interest. Structures mathematics for an engineering audience, with emphasis on engineering applications. New in the Second Edition: Much of the material has been rearranged and revised for pedagogical reasons. The original first chapter has been split in order to allow a more thorough treatment of basic probability before tackling random processes and dynamical systems. The final chapter has been broken into two pieces to provide separate emphasis on process metrics and the ergodic decomposition of affine functionals. Many classic inequalities are now incorporated into the text, along with proofs; and many citations have been added.


Introduction.- Probability Spaces.- Sample Spaces.- Metric Spaces.- Measurable Spaces.- Borel Measurable Spaces.- Polish Spaces.- Probability Spaces.- Complete Probability Spaces.- Extension.- Random Processes and Dynamical Systems.- Measurable Functions and Random Variables.- Approximation of Random Variables and Distributions.- Random Processes and Dynamical Systems.- Distributions.- Equivalent Random Processes.- Codes, Filters, and Factors.- Isomorphism.- Standard Alphabets.- Extension of Probability Measures.- Standard Spaces.- Some Properties of Standard Spaces.- Simple Standard Spaces.- Characterization of Standard Spaces.- Extension in Standard Spaces.- The Kolmogorov Extension Theorem.- Bernoulli Processes.- Discrete B-Processes.- Extension Without a Basis.- Lebesgue Spaces.- Lebesgue Measure on the Real Line.- Standard Borel Spaces.- Products of Polish Spaces.- Subspaces of Polish Spaces.- Polish Schemes.- Product Measures.- IID Random Processes and B-processes.- Standard Spaces vs. Lebesgue Spaces.- Averages.- Discrete Measurements.- Quantization.- Expectation.- Limits.- Inequalities.- Integrating to the Limit.- Time Averages.- Convergence of Random Variables.- Stationary Random Processes.- Block and Asymptotic Stationarity.- Conditional Probability and Expectation.- Measurements and Events.- Restrictions of Measures.- Elementary Conditional Probability.- Projections.- The Radon-Nikodym Theorem.- Probability Densities.- Conditional Probability.- Regular Conditional Probability.- Conditional Expectation.- Independence and Markov Chains.- Ergodic Properties.- Ergodic Properties of Dynamical Systems.- Implications of Ergodic Properties.- Asymptotically Mean Stationary Processes.- Recurrence.- Asymptotic Mean Expectations.- Limiting Sample Averages.- Ergodicity.- Block Ergodic and Totally Ergodic Processes.- The Ergodic Decomposition.- Ergodic Theorems.- The Pointwise Ergodic Theorem.- Mixing Random Processes.- Block AMS Processes.- The Ergodic Decomposition of AMS Systems.- The Subadditive Ergodic Theorem.- Process Approximation and Metrics.- Distributional Distance.- Optimal Coupling Distortion and dp Distance.- Prohorov and Variational Distances.- Evaluating dp .- Measures on Measures.- The Ergodic Decomposition.- The Ergodic Decomposition Revisited.- The Ergodic Decomposition of Markov Processes.- Barycenters.- Affine Functions of Measures.- The Ergodic Decomposition of Affine Functionals.- References.-

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