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 and expectations of hitting times. It also includes a comprehensive bibliography.

Format: Hardback
Length: 401 pages
Publication date: 29 March 2022
Publisher: Springer Nature Switzerland AG

This comprehensive two-volume monograph delves into 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 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 also introduces novel asymptotic recurrent algorithms for phase space reduction. Additionally, the text addresses effective conditions of weak convergence for distributions of hitting times and convergence of expectations of hitting times for regularly and singularly perturbed finite Markov chains and semi-Markov processes. Furthermore, the book includes a comprehensive bibliography of significant works in the field, making it an invaluable resource for both graduate students and theoretical and applied researchers engaged in the study of stochastic processes and their applications.

Introduction:
This book is a comprehensive two-volume monograph dedicated to the exploration of limit and ergodic theorems for a wide range of stochastic processes, including Markov chains, semi-Markov processes, and multi-alternating regenerative processes with semi-Markov modulation. The first volume specifically focuses on the study of 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.

Chapter 1:
In Chapter 1, the authors introduce the necessary and sufficient conditions for weak convergence for first-rare-event times and convergence in the topology J for first-rare-event processes. These conditions are derived based on the concepts of martingales and Markov chains. The chapter also presents new asymptotic recurrent algorithms of phase space reduction, which are crucial for analyzing complex stochastic systems.

Chapter 2:
Chapter 2 delves into the effective conditions of weak convergence for distributions of hitting times. The authors discuss various approaches, including the use of martingale arguments and the theory of stochastic differential equations, to establish these conditions. They also provide examples and illustrations to demonstrate the practical applications of the proposed methods.

Chapter 3:
Chapter 3 focuses on convergence of expectations of hitting times for regularly and singularly perturbed finite Markov chains and semi-Markov processes. The authors introduce the concept of hitting times and discuss the convergence of expectations under different perturbation scenarios. They also introduce the notion of regenerative processes and discuss their applications in various fields, such as biology and finance.

Chapter 4:
Chapter 4 presents a comprehensive bibliography of major works in the field of limit. The bibliography. The authors have compiled a comprehensive list of significant publications, books, and research papers that cover various aspects of stochastic processes and their applications. The bibliography is organized by topic areas and provides an effective reference for both graduate students and theoretical and applied researchers interested in advancing their knowledge in this field.

Conclusion:
In conclusion, this book is a valuable resource for both graduate students and theoretical and applied researchers studying stochastic processes and their applications. The first volume provides essential and sufficient conditions for weak convergence, introduces asymptotic recurrent algorithms for phase space reduction, and discusses effective conditions of weak convergence for distributions of hitting times and convergence of expectations of hitting times. The comprehensive bibliography of major works in the field makes it an invaluable reference for further research and exploration.

Weight: 793g
Dimension: 235 x 155 (mm)
ISBN-13: 9783030924027
Edition number: 1st ed. 2022