Open Access
March 2012 Convergence of recurrence of blocks for mixing processes
Dong Han Kim
Osaka J. Math. 49(1): 1-20 (March 2012).


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.


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Dong Han Kim. "Convergence of recurrence of blocks for mixing processes." Osaka J. Math. 49 (1) 1 - 20, March 2012.


Published: March 2012
First available in Project Euclid: 21 March 2012

zbMATH: 1263.37014
MathSciNet: MR2903251

Primary: 37A50
Secondary: 37M25 , 94A17

Rights: Copyright © 2012 Osaka University and Osaka City University, Departments of Mathematics

Vol.49 • No. 1 • March 2012
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