Limit theorems for numbers of returns in arrays under φ -mixing

We consider a φ-mixing shift T on a sequence space ω and study the number N of returns {TqN(n)ω A na} at times qN(n) to a cylinder Ana constructed by a sequence a ω where n runs either until a fixed integer N or until a time τN of the first return {TqN(n)ω A mb} to another cylinder Amb constructed by b ω. Here, qN(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 in comparison to [20]. Still, the dependence on N requires certain conditions under which we obtain Poisson distributions limits of N when counting is until N as N →∞ and geometric distributions limits when counting is until τN as N →∞. The results and the setup are similar to [17] where multiple returns are considered but under the stronger ψ-mixing assumption.