Perturbed Semi-Markov Type Processes II: Ergodic Theorems for Multi-Alternating Regenerative Processes

This book is a comprehensive 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. It presents a complete classification of ergodic theorems and discusses new asymptotic recurrent algorithms for phase space reduction. It also includes a study of super-long, long, and short time ergodic theorems and a comprehensive bibliography. It is an effective reference for researchers studying stochastic processes and their applications.

Format: Paperback / softback
Length: 413 pages
Publication date: 23 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 second volume, in particular, presents a thorough classification of ergodic theorems for alternating regenerative processes, encompassing over twenty-five such theorems. The text addresses innovative asymptotic recurrent algorithms for phase space reduction in multi-alternating regenerative processes driven by regularly and singularly perturbed finite semi-Markov processes. Additionally, it explores a novel investigation into super-long, long, and short time ergodic theorems for these processes. Furthermore, the book includes an extensive 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 practical 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 stability of complex systems. In this monograph, we focus 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.

Volume 2:
The second volume of this monograph is dedicated to the presentation of a comprehensive classification of ergodic theorems for alternating regenerative processes. These processes are characterized by their ability to switch between different states or modes over time, and they exhibit rich dynamics and complex behavior. The book includes over twenty-five such theorems, covering various aspects of ergodic theory and providing a solid foundation for further research.

Asymptotic Recurrent Algorithms:
One of the key contributions of this volume is the development of new asymptotic recurrent algorithms for phase space reduction in multi-alternating regenerative processes modulating by regularly and singularly perturbed finite semi-Markov processes. Phase space reduction is a crucial technique in analyzing complex systems, as it allows for the simplification of the system's dynamics and the computation of important quantities such as ergodic averages and Lyapunov exponents. The proposed algorithms are based on novel ideas and provide significant improvements over existing methods, making them valuable tools for both theoretical and applied researchers.

Super-Long, Long, and Short Time Ergodic Theorems:
In addition to the classification of ergodic theorems, the book also features a new study of super-long, long, and short time ergodic theorems for alternating regenerative processes. These theorems provide insights into the long-term behavior of the system and are particularly useful in understanding the stability and predictability of complex systems. The study includes both theoretical analysis and numerical simulations, demonstrating the effectiveness of the proposed methods.

Conclusion:
This two-volume monograph is a valuable resource for both graduate students and theoretical and applied researchers interested in the study of limit and ergodic theorems for Markov chains, semi-Markov processes, and multi-alternating regenerative processes with semi-Markov modulation. The comprehensive coverage of ergodic theorems, along with the development of innovative algorithms for phase space reduction, makes it an essential tool for advancing our understanding of these complex systems. The book will undoubtedly inspire further research and applications in various fields, including finance, biology, and engineering.

Weight: 658g
Dimension: 235 x 155 (mm)
ISBN-13: 9783030924010
Edition number: 1st ed. 2022