First-rare-event times often appear in probabilistic research under such names as life time, ruin time, stopping time, death time, catastrophic time, extension time, epidemic time, record time, failure time, killing time, passage time, hitting time, absorption time, transition time, etc. Notion of first-rare-event times is usually used to describe the first occurrence of events which have small probability of appearance. One of the fundamental questions here is question of asymptotic analysis of distributions of first-rare-event times taking into account possible changes of model characteristics. Semi-Markov processes are often used for this kind of modeling. This book is devoted to the analysis of conditions of weak convergence of first-rare-event times for a semi-Markov process with finite set of states in both non-triangular array case and in the series of schemes. Book is oriented on professional researchers working with asymptotic analysis of perturbed random functionals defined on Markov chains and semi-Markov processes.

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