Stationary Processes and Discrete Parameter Markov Processes

This textbook covers the theory of weakly stationary and discrete-time Markov processes,with an emphasis on developing intuition and context through simple examples. It includes coverage of ergodic theory,dynamical systems,the Markov property,birth-death chains,hitting probabilities,absorption,iterated random maps,large deviation theory,geometric rates of convergence,special topics,and applications. The book is designed for a single graduate-level course in probability and features a modular design and engaging exercises.

Format: Hardback
Length: 449 pages
Publication date: 03 December 2022
Publisher: Springer International Publishing AG

This comprehensive textbook delves into two distinct branches of stochastic processes: weakly stationary processes and discrete parameter Markov processes. Starting from simple examples, the authors aim to establish a solid foundation of context and intuition before formalizing the theoretical aspects of each topic. By adopting an inviting approach, the book illuminates the key ideas and computations involved in the proofs, providing an ideal basis for further study.

After briefly reviewing the fundamentals of Fourier analysis, the book begins with an introduction to the spectral representation of stationary processes. This is followed by a comprehensive exploration of ergodic theory, encompassing Birkhoffs Ergodic Theorem and an introduction to dynamical systems. The Markov property is then assumed, and the theory of discrete parameter Markov processes is studied on a general state space. Chapters in the book cover a diverse range of topics, including birth–death chains, hitting probabilities, absorption, the representation of Markov processes as iterates of random maps, and large deviation theory for Markov processes.

A notable chapter focuses on geometric rates of convergence to equilibrium, introducing a splitting condition that uniquely captures the recurrence structure of certain iterated maps. The book concludes with a selection of special topics, including applications of large deviation theory, the FKG inequalities, coupling methods, and the Kalman filter.

Designed with short chapters and a modular structure, this textbook provides an in-depth examination of stationary and discrete-time Markov processes. It is particularly well-suited for students and instructors alike, thanks to its accessible, example-driven approach and engaging exercises throughout. A single, graduate-level course in probability is assumed as a prerequisite for understanding the material covered in this textbook.

Weight: 948g
Dimension: 235 x 155 (mm)
ISBN-13: 9783031009419
Edition number: 1st ed. 2022