Let {X(t):t≥0} be a regenerative process with excursion process {Y(u):0•u<T,T} and state space (S,nS). Let An be a sequence of sets in nS 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<∈, some non-exponential limits are obtained in the null recurrent case. The results are illustrated via the age process.
