Perturbed Semi-Markov Type Processes I: Limit Theorems for Rare-Event Times and Processes

This book is the first of a two-volume monograph on limit and ergodic theorems for regularly and singularly perturbed Markov chains, semi-Markov processes, and multi-alternating regenerative processes with semi-Markov modulation. It discusses weak convergence for first-rare-event times and convergence in the topology J for first-rare-event processes, introduces asymptotic recurrent algorithms of phase space reduction, and covers effective conditions for weak convergence of distributions of hitting times and expectations of hitting times. It also includes a comprehensive bibliography.

Format: Paperback / softback
Length: 401 pages
Publication date: 30 March 2023
Publisher: Springer Nature Switzerland AG

This comprehensive two-volume monograph delves into the study of limit and ergodic theorems for a diverse range of Markov chains, semi-Markov processes, and multi-alternating regenerative processes with semi-Markov modulation. The first volume, spanning over 1000 pages, presents essential 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. It introduces innovative asymptotic recurrent algorithms for phase space reduction, addressing effective conditions of weak convergence for distributions of hitting times and convergence of expectations of hitting times for these chains and processes. Additionally, the book includes a comprehensive bibliography of significant works in the field, serving as an invaluable resource for both graduate students and theoretical and applied researchers engaged in the study of stochastic processes and their applications.

Introduction:
The study of limit and ergodic theorems holds immense importance in the field of stochastic processes, as they provide insights into the behavior and characteristics of complex systems. This monograph focuses on a wide range of Markov chains, semi-Markov processes, and multi-alternating regenerative processes with semi-Markov modulation, which are fundamental models in various applications.

First Volume:
The first volume of this monograph is dedicated to the study of limit and ergodic theorems for regularly and singularly perturbed Markov chains. It begins by introducing the necessary concepts and definitions, including Markov chains, transition probabilities, and ergodic theorems. The authors then present essential 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.

New Asymptotic Recurrent Algorithms:
One of the key contributions of this volume is the introduction of new asymptotic recurrent algorithms for phase space reduction. These algorithms are designed to efficiently compute the limiting distribution of a Markov chain or a semi-Markov process, which is crucial for understanding its behavior and analyzing its properties. The authors discuss the theoretical background and computational aspects of these algorithms, providing practical insights for both graduate students and researchers.

Effective Conditions of Weak Convergence:
In addition to phase space reduction, the book addresses effective conditions of weak convergence for distributions of hitting times. The authors discuss various approaches and methods to establish these conditions, which are essential for analyzing the long-term behavior of Markov chains and semi-Markov processes. They also provide examples and illustrations to demonstrate the practical applications of these results.

Convergence of Expectations of Hitting Times:
The book also delves into the convergence of expectations of hitting times for regularly and singularly perturbed finite Markov chains and semi-Markov processes. The authors discuss the theoretical foundations and methods for computing the expectations of hitting times, including the use of martingales and Markov chains. They also provide examples and applications to illustrate the theoretical concepts and their practical implications.

Comprehensive Bibliography:
The monograph includes a comprehensive bibliography of major works in the field. It serves as a valuable resource for both graduate students and theoretical and applied researchers seeking to expand their knowledge and understanding of stochastic processes and their applications. The bibliography covers a wide range of topics, including Markov chains, semi-Markov processes, ergodic theory, and stochastic differential equations.

Conclusion:
In conclusion, this two-volume monograph is a comprehensive and authoritative resource for the study of limit and ergodic theorems for Markov chains, semi-Markov processes, and multi-alternating regenerative processes with semi-Markov modulation. It provides essential and sufficient conditions for weak convergence, introduces new asymptotic recurrent algorithms for phase space reduction, discusses effective conditions of weak convergence for distributions of hitting times, and explores convergence of expectations of hitting times. The comprehensive bibliography serves as an invaluable resource for both graduate students and researchers. By presenting a wealth of knowledge and insights, this monograph contributes significantly to the field of stochastic processes and their applications.

Weight: 640g
Dimension: 235 x 155 (mm)
ISBN-13: 9783030924058
Edition number: 1st ed. 2022