Limits theorem of first passages times to regenerative processes

Abstract

The current work continues the author’s investigation in the field of extreme values analysis. The investigation is concerned with the first passage time of a level u for a wide class of regenerative random processes. The common statement of the problem in most of works in the field under consideration dealt with cases when time of observation, t, and u tend to infinity. Alternatively, we investigate the case when t tends to infinity, whereas u is a fixed number. We establish a general limit theorem for the first passage time of a level u by a regenerative process. This topic is closely associated with the asymptotic behavior of extreme values of regenerative processes. In proving the main result, we establish an important lemma concerning the asymptotic behavior of probabilities for a class of random sums, which may be of independent interest. Necessity of the study of such sums occurs in many areas: mathematical reliability theory, queuing theory, some statistical physics problems. In addition, the work provides examples of applications of the obtained general results to some problems, which arise in applied areas: model of counters of type Geiger-Muller, estimation of the reliability of a redundant system with recovery, the problem of the first passage time of a level u by queue length in the queuing system M/M/1.