This book is a comprehensive treatment of Markov chains with stationary transition probabilities, focusing on their mathematical properties and applications. It is written by Kai Lai Chung, a professor at Syracuse University. The book is divided into two parts: Part I deals with discrete parameter Markov chains, while Part II addresses continuous parameter Markov chains. The former is on a mathematical level comparable to Feller's Introduction to Probability Theory and Its Applications, Vol. I. Part II requires knowledge of the elementary theory of real functions and some measure-theoretic concepts from Doob's Stochastic Processes. The book presents the theory of Markov chains as a distinct area of study, emphasizing the unique properties of chains with denumerable state spaces. It discusses the strong Markov property, which is always applicable in this context, and the principal limit theorem, which is presented in its final form. The book also addresses the challenges of continuous parameter Markov chains, which are the first essentially discontinuous processes studied in detail. It explores the discontinuities of sample functions and the local properties of transition probability functions, which are central to the theory. The book draws on the work of several mathematicians, including Kolmogorov, Doob, and Levy, and presents new results and insights. It includes historical notes and acknowledgments, and it is written with a modern, rigorous approach to stochastic processes. The book also discusses the theory of "boundaries" in relation to the compactification of denumerable state spaces. It concludes with a discussion of open problems and the need for further research in the field. The book is intended for mathematicians and researchers interested in probability theory and stochastic processes.This book is a comprehensive treatment of Markov chains with stationary transition probabilities, focusing on their mathematical properties and applications. It is written by Kai Lai Chung, a professor at Syracuse University. The book is divided into two parts: Part I deals with discrete parameter Markov chains, while Part II addresses continuous parameter Markov chains. The former is on a mathematical level comparable to Feller's Introduction to Probability Theory and Its Applications, Vol. I. Part II requires knowledge of the elementary theory of real functions and some measure-theoretic concepts from Doob's Stochastic Processes. The book presents the theory of Markov chains as a distinct area of study, emphasizing the unique properties of chains with denumerable state spaces. It discusses the strong Markov property, which is always applicable in this context, and the principal limit theorem, which is presented in its final form. The book also addresses the challenges of continuous parameter Markov chains, which are the first essentially discontinuous processes studied in detail. It explores the discontinuities of sample functions and the local properties of transition probability functions, which are central to the theory. The book draws on the work of several mathematicians, including Kolmogorov, Doob, and Levy, and presents new results and insights. It includes historical notes and acknowledgments, and it is written with a modern, rigorous approach to stochastic processes. The book also discusses the theory of "boundaries" in relation to the compactification of denumerable state spaces. It concludes with a discussion of open problems and the need for further research in the field. The book is intended for mathematicians and researchers interested in probability theory and stochastic processes.