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

Abstract

We consider a [Formula: see text]-mixing shift [Formula: see text] on a sequence space [Formula: see text] and study the number [Formula: see text] of returns [Formula: see text] at times [Formula: see text] to a cylinder [Formula: see text] constructed by a sequence [Formula: see text] where [Formula: see text] runs either until a fixed integer [Formula: see text] or until a time [Formula: see text] of the first return [Formula: see text] to another cylinder [Formula: see text] constructed by [Formula: see text]. Here, [Formula: see text] are certain functions of [Formula: see text] taking on nonnegative integer values when [Formula: see text] runs from 0 to [Formula: see text] and the dependence on [Formula: see text] is the main generalization here in comparison to [20]. Still, the dependence on [Formula: see text] requires certain conditions under which we obtain Poisson distributions limits of [Formula: see text] when counting is until [Formula: see text] as [Formula: see text] and geometric distributions limits when counting is until [Formula: see text] as [Formula: see text]. The results and the setup are similar to [17] where multiple returns are considered but under the stronger [Formula: see text]-mixing assumption.