Probability, Random Processes, and Ergodic Properties
This book is a self-contained treatment of the theory of probability, random processes. It is intended to lay solid theoretical foundations for advanced probability, that is, for measure and integration theory, and to develop in depth the long term time average behavior of measurements made on random processes with general output alphabets. Unlike virtually all texts on the topic, considerable space is devoted to processes that violate the usual assumptions of stationarity and ergodicity, yet which still possess the fundamental properties of convergence of long term averages to appropriate expectations. The theory of asymtotically mean stationary processes and the ergodic decomposition are both treated in depth for both one-sided and two-sided random processes. In addition, the book treats many of the fundamental results such as the Kolmogorov extension theorem and the ergodic decomposition theorem. Much of the material has not previously appeared in book form, and the treatment takes advantage of many recent generalizations and simplifications.
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PROBABILITY AND RANDOM PROCESSES
SPACES OF MEASURES
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alphabet basis binary sequence Borel sets Borel space bounded measurements class of measurements completes the proof conditional expectation conditional probability consider construction contains convergence Corollary countable extension property countably additive defined definition denote discrete measurements disjoint distribution dynamical system equivalent ergodic decomposition ergodic properties ergodic theorem events F example exists field F finite fields finitely additive G-measurable implies index set indicator functions inequality invariant events invariant measurements inverse images isomorphic lim sup limiting sample averages mapping measurable function measurable space measurable with respect metric space nonempty nonnegative open sets open sphere output Polish space probability measure probability space product space properties with respect prove random process random variables rectangles recurrent result sequence space set F shift simple functions standard spaces stationary and ergodic stationary mean stationary measures sub-o-field subadditive subsets subspace Suppose theory thin cylinders uniformly integrable vectors yield