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 providesan effective reference for both graduate students as well as theoretical and applied researchers studying stochastic processes and their applications.

Autorentext
Dmitrii Silvestrov: Candidate of Science [PhD], (1969, Kiev University), Doctor of Science (1972, Kiev University). Professor at Kiev University (1974-1992), Mälardalen University (1999-2009) and Stockholm University from 2009. At present, professor emeritus at Stockholm and Mälardalen Universities. The main areas of research is Stochastic Processes and their Applications. The author of 13 books and more than 170 research papers. Supervised 22 doctoral students who subsequently obtained PhD degrees.

Inhalt
Preface.- List of symbols.- Introduction.- Part I: First-Rare-Event Times for Regularly Perturbed Semi-Markov Processes.- Flows of Rare Events for Regularly Perturbed Semi-Markov Processes.- Generalizations of Limit Theorems for First-Rare-Event Times.- First-Rare-Event Times for Perturbed Risk Processes.- First-Rare-Event Times for Perturbed Closed Queuing Systems.- First-Rare-Event Times for Perturbed M/M-Type Queuing Systems.- Part II: Hitting Times and Phase Space Reduction for Perturbed Semi-Markov Processes.- Asymptotically Comparable Functions.- Perturbed Semi-Markov Processes and Reduction of Phase Space.- Asymptotics of Hitting Times for Perturbed Semi-Markov Processes.- Asymptotics for Expectations of Hitting Times for Perturbed Semi-Markov Processes.- Generalizations and Examples of Limit Theorems for Hitting Times.- Limit Theorems for Randomly Stopped Stochastic Processes.- Methodological and Bibliographical Notes.- References.- Index.