Convergence of recurrence of blocks for mixing processes (Q411701)

It is known that convergence of the logarithm of the first-return time (recurrence time) of the initial block normalized by the block length has been investigated in relation to entropy-estimation or data-compression methods such as the Ziv-Lempel algorithm.NEWLINENEWLINE Let \(\{X_n/n\in\mathbb{N}\}\) be a stationary ergodic process on the space of infinite sequences \(({\mathcal A}^{\mathbb{N}},\Sigma,\mathbb{P})\), where \({\mathcal A}\) is a finite set, \(\Sigma\) is the \(\sigma\)-field generated by finite-dimensional cylinders, \(\mathbb{P}\) is a shift-invariant ergodic probability measure and let define by \(R_n\) the first-return time of the initial \(n\)-block \(x^n_1= x_1\cdots x_n\), that is, NEWLINE\[NEWLINER_n(x):= \min\{j\geq 1; x^n_1= x^{j+n}_{j+1}\}.NEWLINE\]NEWLINE The aim of the present paper is to study sharp bounds for the convergence of \(R_n(x) P_n(x)\) to exponential distribution, for mixing processes, where \(P_n(x)\) is the probability of \(x_1\cdots x_n\). Furthermore the author determines the limit of the mean of \(\log(R_n(x) P_n(x))\), as a corollary. In particular, for exponentially \(\varphi\)-mixing processes, the author shows that \(-E[\log(R_n(x) P_n(x))]\) converges exponentially to the Euler's constant, and proves a similar result for the hitting time.