Rare events for stationary processes

Keilson (1979, Markov Chain Models - Rarity and Exponentiality, Springer, N ew York) and Aldous (1989, Probability approximations via the Poisson Clump ing Heuristic, Springer, New York) have given expressions for the asymptoti cs of the mean time until a rare event occurs. Here we extend these results beyond the Markovian setting using the theory for stationary paint process es. We introduce two notions of asymptotic exponentiality in variance and a symptotic independence and we study their implications on the asymptotics o f the mean value of this hitting time under various initial probability mea sures. (C) 2000 Elsevier Science B.V. All rights reserved. MSC. Primary 60G55; secondary 60K25.