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.
Preface.- Introduction.- Summary of Ergodic Theorems for Regenerative
Processes.- Modifications of Hitting Times.- Birth-Death-Type Processes.-
Semi-Markov Processes with Discrete State Spaces and Embedded Regenerative
Processes.- Ergodic Theorems for Queuing Systems.- Semi-Markov Processes with
General State Spaces with Atoms.- Semi-Markov Processes with General State
Spaces and Distributional Atoms.- Semi-Markov Processes with General State
Spaces and One-Step Artificial Regeneration.- Semi-Markov Processes with
General State Spaces and Multi-Step Artificial Regeneration.-
Multi-Alternating Regenerative Processes with Semi-Markov Modulation.-
Multi-Alternating Regenerative Processes Modulating by Uniformly Recurrent
Semi-Markov Processes.- Appendix A. Methodological and Bibliographical
Notes.- References.- Index.
Dmitrii Silvestrov graduated from Kiev University (1968, Mathematics), Candidate of Science [ PhD], (1969, Mathematical Statistics), and Doctor of Science (1972, Mathematical Statistics). Awarded the Prize of the Moscow Mathematical Society (1973) and the Ukrainian Ostrovsky Prize (1977) for work on stochastic processes. Lecturer and Senior Lecturer (1970-1974), Professor (1974-1992, Department of Probability and Mathematical Statistics) and Head of the Statistical Research Centre (1980-1990) at Kiev University. Guest scientist at Umeå University (1991-1992), Senior lecturer at Luleå University of Technology (1992-1994) and at Umeå University (1994-1999). Visiting professor at the Hebrew University of Jerusalem (1993), University of Turku (1998), and University of Rome "La Sapienza" (2015). Professor at the Mälardalen University from 1999 (Emeritus Professor from 2012) and Stockholm University from 2009 (Emeritus Professor from 2016). Member of the editorial boards of the journals "Theory of Probability and Mathematical Statistics" and "Theory of Stochastic Processes". Coordinator of the four EU Tempus Projects. The main research areas are stochastic processes, actuarial and financial mathematics, and statistical software. Author of 13 books and more than 170 research papers. Supervised 22 doctoral students who subsequently obtained PhD degrees.
