Probability, Random Processes, and Ergodic Properties
Gray, Robert M.
2nd ed. 2009, XXXV, 322 p.
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Offers a complete tour of the book and guidelines for its use in the Introduction, allowing readers to see at a glance the topics of interest
Structures the mathematics for an engineering audience with proofs and interpretations from the point of view of signal modeling, processing and coding
Original first chapter has been split in order to allow a more thorough treatment of basic probability before tackling the 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 functional
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.
Second edition of classic text
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.
Completion of event spaces and probability measures is treated in more detail.
More specific examples of random processes have been introduced.
Many classic inequalities are now incorporated into the text, along with proofs; and many citations have been added.
From the Author’s Preface...
This book has a long history. It began over two decades ago as the first half of a book on information and ergodic theory. The intent was and remains to provide a reasonably self-contained advanced (at least for engineers) treatment of measure theory, probability theory, and random processes, with an emphasis on general alphabets and on ergodic and stationary properties of random processes that might be neither ergodic nor stationary.
The intended audience was mathematically inclined engineers who had not had formal courses in measure theoretic probability or ergodic theory. Much of the material is familiar stuff for mathematicians, but many of the topics and results had not then previously appeared in books. The original project grew too large and the first part contained much that would likely bore mathematicians and discourage them from the second part. Hence I finally followed a suggestion to separate the material and split the project in two. The resulting manuscript fills a unique hole in the literature. Personal experience indicates that the intended audience rarely has the time to take a complete course in measure and probability theory in a mathematics or statistics department, at least not before they need some of the material in their research.
I intended in this book to provide a catalogue of many results that I have found need of in my own research together with proofs that I could follow. I also intended to clarify various connections that I had found confusing or insufficiently treated in my own reading. If the book provides similar service for others, it will have succeeded.
Content Level »Research
Keywords »Markov - Signal - communication - information - linear optimization - metrics - signal processing
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.-