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

We consider a $\varphi$-mixing shift $T$ on a sequence space $\text{Om}$ and study the number ${\text{cN}}_N$ of returns $\{T^{q_N(n)}\text{om}\in A_n^a\}$ at times $q_N(n)$ to a cylinder $A_n^a$ constructed by a sequence $a\in \text{Om}$ where $n$ runs either until a fixed integer $N$ or until a time ${\tau }_N$ of the first return ${ T^{q_N(n)}\om\in A^b_m}$toanothercylinder$A^b_m$constructedby$b\in\Om$. 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 ${\text{cN}}_N$ when counting is until $N$ as $N\to \mathrm{\infty }$ and geometric distributions limits when counting is until ${\tau }_N$ as $N\to \mathrm{\infty }$.