Overview
- 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
Part of the book series: Graduate Texts in Mathematics (GTM, volume 293)
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Table of contents (25 chapters)
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Front Matter
Keywords
- 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
- advanced probability textbook
About this book
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.
Authors and Affiliations
About the authors
Rabi Bhattacharya is Professor of Mathematics at The University of Arizona. He is a Fellow of the Institute of Mathematical Statistics and a recipient of the U.S. Senior Scientist Humboldt Award and of a Guggenheim Fellowship. He has made significant contributions to the theory and application of Markov processes, and more recently, nonparametric statistical inference on manifolds. He has served on editorial boards of many international journals and has published several research monographs and graduate texts on probability and statistics.
Edward C. Waymire is Emeritus Professor of Mathematics at Oregon State University. He received a PhD in mathematics from the University of Arizona in the theory of interacting particle systems. His primary research concerns applications of probability and stochastic processes to problems of contemporary applied mathematics pertaining to various types of flows, dispersion, and random disorder. He is a former chief editor of the Annals of Applied Probability, and past president of the Bernoulli Society for Mathematical Statistics and Probability.
Both authors have co-authored numerous books, including A Basic Course in Probability Theory, which is an ideal companion to the current volume.
Bibliographic Information
Book Title: Stationary Processes and Discrete Parameter Markov Processes
Authors: Rabi Bhattacharya, Edward C. Waymire
Series Title: Graduate Texts in Mathematics
DOI: https://doi.org/10.1007/978-3-031-00943-3
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer Nature Switzerland AG 2022
Hardcover ISBN: 978-3-031-00941-9Published: 03 December 2022
eBook ISBN: 978-3-031-00943-3Published: 01 December 2022
Series ISSN: 0072-5285
Series E-ISSN: 2197-5612
Edition Number: 1
Number of Pages: XVII, 449
Number of Illustrations: 5 b/w illustrations
Topics: Probability Theory and Stochastic Processes, Statistics and Computing/Statistics Programs