Renewal theorems in a periodic environment

We study a renewal problem within a periodic environment, departing from the classical renewal theory by relaxing the assumption of independent and identically distributed inter-arrival times. Instead, the conditional distribution of the next arrival time, given the current one, is governed by a periodic kernel, denoted as H. The periodicity property of H is expressed as ℙ(T_k+1 > t   |  T_k) = H(t, T_k), where H(t+T,s+T) = H(t, s). For a fixed time t, we define N_t as the count of events occurring up to time t. The focus is on two temporal aspects: Y_t, the time elapsed since the last event, and X_t, the time until the next event occurs, given by Y_t = t - T_N_t and X_t = T_N_t+1 - t. The study explores the long-term behavior of the distributions of X_t and Y_t.