Dmitrii Silvestrov

Ergodic theorems are a cornerstone of the theory of stochastic processes and their applications. This volume delves into ergodic theorems with explicit power and exponential upper bounds for convergence rates, focusing on Markov chains, renewal processes, and regenerative processes. The book offers a powerful and constructive probabilistic framework by employing the elegant coupling method in conjunction with test functions. Theoretical findings are illustrated with applications to perturbed stochastic networks, alternating Markov processes, risk processes, quasi-stationary distributions, and the renewal theorem, all of which feature explicit convergence rate bounds. Many results presented here are groundbreaking, appearing in publication for the first time. This is the first volume of a two-volume monograph dedicated to ergodic theorems. While this volume centers on Markovian and regenerative models, the second volume extends the scope to semi-Markov processes and multi-alternating regenerative processes with semi-Markov modulation. Designed with researchers and advanced students in mind, the content is thoughtfully structured by complexity, making it suitable for self-study or as a resource for upper-level coursework. Each chapter is self-contained and complemented by a comprehensive bibliography, ensuring its value as a long-lasting reference. An essential resource for theoretical and applied research, this book significantly contributes to the field of stochastic processes and will remain a key reference for years to come.

Ergodic theorems are a cornerstone of the theory of stochastic processes and their applications. This book is the second volume of a two-volume monograph dedicated to ergodic theorems. While the first volume centers on Markovian and regenerative models, the second volume extends the scope to semi-Markov processes and multi-alternating regenerative processes with semi-Markov modulation and delves into ergodic theorems with explicit power and exponential upper bounds for convergence rates for such processes. The book offers a powerful and constructive probabilistic framework by employing coupling ergodic theorems presented in the first volume in conjunction with the method of artificial regeneration and test functions. Theoretical findings are illustrated with applications to semi-Markov Monte Carlo algorithms and perturbed queuing systems featuring explicit convergence rate bounds. Many results presented in the book are groundbreaking, appearing in publication for the first time. Designed with researchers and advanced students in mind, the content is thoughtfully structured by complexity, making it suitable for self-study or as a resource for upper-level coursework. Each chapter is self-contained and complemented by a comprehensive bibliography, ensuring its value as a long-lasting reference. An essential resource for theoretical and applied research, this book significantly contributes to the field of stochastic processes and will remain a key reference for years to come.

This book is the first volume of a two-volume monograph devoted to the study of limit and ergodic theorems for regularly and singularly perturbed Markov chains, semi-Markov processes, and multi-alternating regenerative processes with semi-Markov modulation. The first volume presents necessary and sufficient conditions for weak convergence for first-rare-event times and convergence in the topology J for first-rare-event processes defined on regularly perturbed finite Markov chains and semi-Markov processes. The text introduces new asymptotic recurrent algorithms of phase space reduction. It also addresses both effective conditions of weak convergence for distributions of hitting times as well as convergence of expectations of hitting times for regularly and singularly perturbed finite Markov chains and semi-Markov processes. The book also contains a comprehensive bibliography of major works in the field. It provides an effective reference for both graduate students as well as theoretical and applied researchers studying stochastic processes and their applications.

This book is the second volume of a two-volume monograph devoted to the study of limit and ergodic theorems for regularly and singularly perturbed Markov chains, semi-Markov processes, and multi-alternating regenerative processes with semi-Markov modulation. The second volume presents a complete classification of ergodic theorems for alternating regenerative processes, including more than twenty-five such theorems. The text addresses new asymptotic recurrent algorithms of phase space reduction for multi-alternating regenerative processes modulating by regularly and singularly perturbed finite semi-Markov processes. It also features a new study of super-long, long, and short time ergodic theorems for these processes.The book also contains a comprehensive bibliography of major works in the field. It provides an effective reference for both graduate students as well as theoretical and applied researchers studying stochastic processes and their applications.

This book is the first volume of a two-volume monograph devoted to the study of limit and ergodic theorems for regularly and singularly perturbed Markov chains, semi-Markov processes, and multi-alternating regenerative processes with semi-Markov modulation. The first volume presents necessary and sufficient conditions for weak convergence for first-rare-event times and convergence in the topology J for first-rare-event processes defined on regularly perturbed finite Markov chains and semi-Markov processes. The text introduces new asymptotic recurrent algorithms of phase space reduction. It also addresses both effective conditions of weak convergence for distributions of hitting times as well as convergence of expectations of hitting times for regularly and singularly perturbed finite Markov chains and semi-Markov processes. The book also contains a comprehensive bibliography of major works in the field. It provides an effective reference for both graduate students as well as theoretical and applied researchers studying stochastic processes and their applications.

This book is the second volume of a two-volume monograph devoted to the study of limit and ergodic theorems for regularly and singularly perturbed Markov chains, semi-Markov processes, and multi-alternating regenerative processes with semi-Markov modulation. The second volume presents a complete classification of ergodic theorems for alternating regenerative processes, including more than twenty-five such theorems. The text addresses new asymptotic recurrent algorithms of phase space reduction for multi-alternating regenerative processes modulating by regularly and singularly perturbed finite semi-Markov processes. It also features a new study of super-long, long, and short time ergodic theorems for these processes.The book also contains a comprehensive bibliography of major works in the field. It provides an effective reference for both graduate students as well as theoretical and applied researchers studying stochastic processes and their applications.

The book presents new methods of asymptotic analysis for nonlinearly perturbed semi-Markov processes with a finite phase space. These methods are based on special time-space screening procedures for sequential phase space reduction of semi-Markov processes combined with the systematical use of operational calculus for Laurent asymptotic expansions. Effective recurrent algorithms are composed for getting asymptotic expansions, without and with explicit upper bounds for remainders, for power moments of hitting times, stationary and conditional quasi-stationary distributions for nonlinearly perturbed semi-Markov processes. These results are illustrated by asymptotic expansions for birth-death-type semi-Markov processes, which play an important role in various applications. The book will be a useful contribution to the continuing intensive studies in the area. It is an essential reference for theoretical and applied researchersin the field of stochastic processes and their applications that will contribute to continuing extensive studies in the area and remain relevant for years to come.