Stationary processes and discrete parameter Markov processes / Rabi Bhattacharya, Edward C. Waymire.
Bhattacharya, R. N. (Rabindra Nath), 1937- author.; Waymire, Edward C., author.
2022
QA274
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Title Stationary processes and discrete parameter Markov processes / Rabi Bhattacharya, Edward C. Waymire.
Author Bhattacharya, R. N. (Rabindra Nath), 1937- author.
ISBN 9783031009433 (electronic bk.)
3031009436 (electronic bk.)
303100941X
9783031009419
DOI https://doi.org/10.1007/978-3-031-00943-3
Imprint Cham, Switzerland : Springer, 2022.
Language English
Description 1 online resource
Other Standard Identifiers 10.1007/978-3-031-00943-3 doi
Call Number QA274
Dewey Decimal Classification 519.2/3
Summary 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 Birkhoffs 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 birthdeath 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.
Access Note Access limited to authorized users.
Source of Description Online resource; title from PDF title page (Springerlink, viewed December 8, 2022).
Added Author Waymire, Edward C., author.
Series Graduate texts in mathematics.
Available in Other Form Print version: 9783031009419
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Symbol Definition List
1. Fourier Analysis: A Brief
2. Weakly Stationary Processes and their Spectral Measures
3. Spectral Representation of Stationary Processes
4. Birkhoffs Ergodic Theorem
5. Subadditive Ergodic Theory
6. An Introduction to Dynamical Systems
7. Markov Chains
8. Markov Processes with General State Space
9. Stopping Times and the Strong Markov Property
10. Transience and Recurrence of Markov Chains
11. BirthDeath Chains
12. Hitting Probabilities & Absorption
13. Law of Large Numbers and Invariant Probability for Markov Chains by Renewal Decomposition
14. The Central Limit Theorem for Markov Chains by Renewal Decomposition
15. Martingale Central Limit Theorem
16. Stationary Ergodic Markov Processes: SLLN & FCLT
17. Linear Markov Processes
18. Markov Processes Generated by Iterations of I.I.D. Maps
19. A Splitting Condition and Geometric Rates of Convergence to Equilibrium
20. Irreducibility and Harris Recurrent Markov Processes
21. An Extended PerronFrobenius Theorem and Large Deviation Theory for Markov Processes
22. Special Topic: Applications of Large Deviation Theory
23. Special Topic: Associated Random Fields, Positive Dependence, FKG Inequalities
24. Special Topic: More on Coupling Methods and Applications
25. Special Topic: An Introduction to Kalman Filter
A. Spectral Theorem for Compact Self-Adjoint Operators and Mercers Theorem
B. Spectral Theorem for Bounded Self-Adjoint Operators
C. Borel Equivalence for Polish Spaces
D. HahnBanach, Separation, and Representation Theorems in Functional Analysis
References
Author Index
Subject Index.

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