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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.
Symbol Definition List
Fourier Analysis: A Brief Survey
Weakly Stationary Processes and Their Spectral Measures
Spectral Representation of Stationary Processes
Birkhoff's Ergodic Theorem
Subadditive Ergodic Theory
An Introduction to Dynamical Systems
Markov Chains
Markov Processes with General State Space
Stopping Times and the Strong Markov Property
Transience and Recurrence of Markov Chains
Birth–Death Chains
Hitting Probabilities &amp Absorption
Law of Large Numbers and Invariant Probability for Markov Chains by Renewal Decomposition
The Central Limit Theorem for Markov Chains by Renewal Decomposition
Martingale Central Limit Theorem
Stationary Ergodic Markov Processes: SLLN &amp FCLT
Linear Markov Processes
Markov Processes Generated by Iterations of I.I.D. Maps
A Splitting Condition and Geometric Rates of Convergence to Equilibrium
Irreducibility and Harris Recurrent Markov Processes
An Extended Perron–Frobenius Theorem and Large Deviation Theory for Markov Processes
Special Topic: Applications of Large Deviation Theory
Special Topic: Associated Random Fields, Positive Dependence, FKG Inequalities
Special Topic: More on Coupling Methods and Applications
Special Topic: An Introduction to Kalman Filter
Spectral Theorem for Compact Self-Adjoint Operators and Mercer's Theorem
Spectral Theorem for Bounded Self-Adjoint Operators
Borel Equivalence for Polish Spaces
Hahn–Banach, Separation, and Representation Theorems in Functional Analysis
Related Textbooks and Monographs
Author Index
Subject Index