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.
Author(s): Rabi Bhattacharya, Edward C. Waymire
Series: Graduate Texts in Mathematics, 293
Publisher: Springer
Year: 2022
Language: English
Pages: 448
City: Cham
Tags: Markov Processe, Weakly Stationary Processes, Spectral Representation, Ergodic Theory, Markov Chains, Large Deviations
Preface
Contents
Symbol Definition List
1 Fourier Analysis: A Brief Survey
Exercises
2 Weakly Stationary Processes and Their Spectral Measures
Exercises
3 Spectral Representation of Stationary Processes
Exercises
4 Birkhoff's Ergodic Theorem
Exercises
5 Subadditive Ergodic Theory
Exercises
6 An Introduction to Dynamical Systems
Exercises
7 Markov Chains
Exercises
8 Markov Processes with General State Space
Exercises
9 Stopping Times and the Strong Markov Property
Exercises
10 Transience and Recurrence of Markov Chains
Exercises
11 Birth–Death Chains
Exercises
12 Hitting Probabilities & Absorption
Exercises
13 Law of Large Numbers and Invariant Probability for Markov Chains by Renewal Decomposition
Exercises
14 The Central Limit Theorem for Markov Chains by Renewal Decomposition
Exercises
15 Martingale Central Limit Theorem
Exercises
16 Stationary Ergodic Markov Processes: SLLN & FCLT
Exercises
17 Linear Markov Processes
Exercises
18 Markov Processes Generated by Iterations of I.I.D. Maps
Exercises
19 A Splitting Condition and Geometric Rates of Convergence to Equilibrium
Exercises
20 Irreducibility and Harris Recurrent Markov Processes
Exercises
21 An Extended Perron–Frobenius Theorem and Large Deviation Theory for Markov Processes
Exercises
22 Special Topic: Applications of Large Deviation Theory
Exercises
23 Special Topic: Associated Random Fields, Positive Dependence, FKG Inequalities
Exercises
24 Special Topic: More on Coupling Methods and Applications
Exercises
25 Special Topic: An Introduction to Kalman Filter
Exercises
A Spectral Theorem for Compact Self-Adjoint Operators and Mercer's Theorem
B Spectral Theorem for Bounded Self-Adjoint Operators
C Borel Equivalence for Polish Spaces
D Hahn–Banach, Separation, and Representation Theorems in Functional Analysis
References
Related Textbooks and Monographs
Author Index
Subject Index
