Convergence of recurrence of blocks for mixing processes

Abstract

Let $R_{n} (x)$ be the first return time of the initial sequence $x_{1} \cdots x_{n}$ of $x = x_{1} x_{2} \cdots$. 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 $x_{1} \cdots x_{n}$. As a corollary, the limit of the mean of $\log(R_{n}(x) P_{n}(x))$ is obtained. For exponentially $\phi$-mixing processes, $-E[\log(R_{n} P_{n})]$ converges exponentially to the Euler's constant. A similar result is observed for the hitting time.