## Osaka Journal of Mathematics

### Convergence of recurrence of blocks for mixing processes

Dong Han Kim

#### 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.

#### Article information

Source
Osaka J. Math., Volume 49, Number 1 (2012), 1-20.

Dates
First available in Project Euclid: 21 March 2012

https://projecteuclid.org/euclid.ojm/1332337235

Mathematical Reviews number (MathSciNet)
MR2903251

Zentralblatt MATH identifier
1263.37014

#### Citation

Kim, Dong Han. Convergence of recurrence of blocks for mixing processes. Osaka J. Math. 49 (2012), no. 1, 1--20. https://projecteuclid.org/euclid.ojm/1332337235

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