Stationary Processes and Discrete Parameter Markov Processes

Description

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. 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