Stationary Processes and Discrete Parameter Markov Processes
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Beschreibung
This textbook explores two distinct stochastic processes that evolve at random: weakly stationary processes and discrete parameter Markov processes. Building from simple examples, the authors focus on developing context and intuition before formalizing the theory of each topic. This inviting approach illuminates the key ideas and computations in the proofs, forming an ideal basis for further study.
After recapping the essentials from Fourier analysis, the book begins with an introduction to the spectral representation of a stationary process. Topics in ergodic theory follow, including Birkhoff’s Ergodic Theorem and an introduction to dynamical systems. From here, the Markov property is assumed and the theory of discrete parameter Markov processes is explored on a general state space. Chapters cover a variety of topics, including birth–death chains, hitting probabilities and absorption, the representation of Markov processes as iterates of random maps, and large deviation theory for Markov processes. A chapter on geometric rates of convergence to equilibrium includes a splitting condition that captures the recurrence structure of certain iterated maps in a novel way. A selection of special topics concludes the book, including applications of large deviation theory, the FKG inequalities, coupling methods, and the Kalman filter.
Featuring many short chapters and a modular design, this textbook offers an in-depth study of stationary and discrete-time Markov processes. Students and instructors alike will appreciate the accessible, example-driven approach and engaging exercises throughout. A single, graduate-level course in probability is assumed.
This textbook explores two distinct stochastic processes that evolve at random: weakly stationary processes and discrete parameter Markov processes. Building from simple examples, the authors focus on developing context and intuition before formalizing the theory of each topic. This inviting approach illuminates the key ideas and computations in the proofs, forming an ideal basis for further study.
After recapping the essentials from Fourier analysis, the book begins with an introduction to the spectral representation of a stationary process. Topics in ergodic theory follow, including Birkhoff’s Ergodic Theorem and an introduction to dynamical systems. From here, the Markov property is assumed and the theory of discrete parameter Markov processes is explored on a general state space. Chapters cover a variety of topics, including birth–death chains, hitting probabilities and absorption, the representation of Markov processes as iterates of random maps, and large deviation theory for Markov processes. A chapter on geometric rates of convergence to equilibrium includes a splitting condition that captures the recurrence structure of certain iterated maps in a novel way. A selection of special topics concludes the book, including applications of large deviation theory, the FKG inequalities, coupling methods, and the Kalman filter.
Featuring many short chapters and a modular design, this textbook offers an in-depth study of stationary and discrete-time Markov processes. Students and instructors alike will appreciate the accessible, example-driven approach and engaging exercises throughout. A single, graduate-level course in probability is assumed.
Offers an in-depth study of stationary and discrete-time Markov processes Builds from simple examples to formal proofs, illuminating key ideas and computations Explores special topics that include applications of large deviation theory, coupling methods, and the Kalman filter
Autor*in
Rabi Bhattacharya
Themen in »Stationary Processes and Discrete Parameter Markov Processes«
weakly stationary processes discrete parameter Markov processes spectral representation of a stationary process Birkhoff’s Ergodic Theorem discrete parameter Markov processes on a general state space birth–death chains hitting probabilities and absorption large deviation theory for Markov processes stationary ergodic Markov processes geometric rates of convergence to equilibrium applications of large deviation theory Extended Perron-Frobenius Theorem FKG inequalities Kalman filter stochastic processes textbook
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"The book is an advanced level measure theoretic probability book. ... The book is an impressive presentation of material, including a huge variety of topics in probability. Because of the wealth of subjects, there is an abundance of possible research topics waiting to challenge new (or experienced) probability experts. This book would be an excellent text for an advanced probability course, and is certainly a valuable reference for those interested in the exciting field of probability." (Myron Hlynka, Mathematical Reviews, March, 2025)
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