Limit theorems for numbers of returns in arrays under $\phi$-mixing

Abstract

We consider a $\phi$-mixing shift $T$ on a sequence space $\Om$ and study the number $\cN_N$ of returns $\{ T^{q_N(n)}\om\in A^a_n\}$ at times $q_N(n)$ to a cylinder $A^a_n$ constructed by a sequence $a\in\Om$ where $n$ runs either until a fixed integer $N$ or until a time $\tau_N$ of the first return $\{ T^{q_N(n)}\om\in A^b_m\}$ to another cylinder $A^b_m$ constructed by $b\in\Om$. Here $q_N(n)$ are certain functions of $n$ taking on nonnegative integer values when $n$ runs from 0 to $N$ and the dependence on $N$ is the main generalization here which requires certain conditions under which we obtain Poisson distributions limits of $\cN_N$ when counting is until $N$ as $N\to\infty$ and geometric distributions limits when counting is until $\tau_N$ as $N\to\infty$.