Hitting and returning to rare events for all alpha-mixing processes

We prove that for any α -mixing stationary process the hitting time of any n -string A n converges, when suitably normalized, to an exponential law. We identify the normalization constant λ ( A n ) . A similar statement holds also for the return time. ablish this result we prove two other results of independent interest. First, we show a relation between the rescaled hitting time and the rescaled return time, generalizing a theorem of Haydn, Lacroix and Vaienti. Second, we show that for positive entropy systems, the probability of observing any n -string in n consecutive observations goes to zero as n goes to infinity.