Rare events for stationary processes

Keilson (1979, Markov Chain Models — Rarity and Exponentiality, Springer, New York) and Aldous (1989, Probability approximations via the Poisson Clumping Heuristic, Springer, New York) have given expressions for the asymptotics of the mean time until a rare event occurs. Here we extend these results beyond the Markovian setting using the theory for stationary point processes. We introduce two notions of asymptotic exponentiality in variance and asymptotic independence and we study their implications on the asymptotics of the mean value of this hitting time under various initial probability measures.