Convergence of recurrence of blocks for mixing processes

Let R _n(x) be the first return time of the initial sequence x ₁...x _n of x = x ₁x ₂.... For mixing processes, sharp bounds for the convergence of R _n(x)P _n(x) to exponential distribution are presented, where P _n(x) is the probability of x ₁...x _n. As a corollary, the limit of the mean of log(R _n(x)P _n(x)) is obtained. For exponentially φ-mixing processes, -E[log(R _n P _n)] converges exponentially to the Euler's constant. A similar result is observed for the hitting time.