Osaka Journal of Mathematics

Convergence of recurrence of blocks for mixing processes

Dong Han Kim

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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

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

First available in Project Euclid: 21 March 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37A50: Relations with probability theory and stochastic processes [See also 60Fxx and 60G10]
Secondary: 94A17: Measures of information, entropy 37M25: Computational methods for ergodic theory (approximation of invariant measures, computation of Lyapunov exponents, entropy)


Kim, Dong Han. Convergence of recurrence of blocks for mixing processes. Osaka J. Math. 49 (2012), no. 1, 1--20.

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