Hitting times of decreasing sets for regenerative processes

Abstract

Let {X(t): t ≧ 0} be a regenerative process with excursion process {Y(u): 0 ≦ u < T, T} and state space . Let An be a sequence of sets in such that pn=P(Y(u)∊An for some 0≦ u < T) → 0. Let Vn be the hitting time of An for the process X. This paper gives a variety of conditions on the excursion process Y to obtain limit theorems for Vn. Apart from obtaining an exponential limit in the positive recurrent case, i.e. ET < x, some non-exponential limits are obtained in the null recurrent case. The results are illustrated via the age process.