Limit theorems for numbers of multiple returns in nonconventional arrays

TL;DR

This paper establishes limit theorems for the number of multiple returns in nonconventional arrays of a $$-mixing process, revealing Poisson and geometric distribution limits under certain conditions, with applications to dynamical systems.

Contribution

It introduces a novel dependence of the functions $q_{i,N}$ on both $n$ and $N$, and derives new limit theorems for multiple return counts in nonconventional arrays and dynamical systems.

Findings

Poisson distribution limits for counts until N

Geometric distribution limits for counts until first return

Results extend to dynamical systems with $$-mixing shifts

Abstract

For a $\psi$-mixing process $\xi_0,\xi_1,\xi_2,...$ we consider the number $\mathcal{N}_N$ of multiple returns $\{\xi_{q_{i,N}(n)}\in\Gamma_N,\, i=1,...,\ell\}$ to a set $\Gamma_N$ for $n$ until either a fixed number $N$ or until the moment $\tau_N$ when another multiple return $\{\xi_{q_{i,N}(n)}\in\Delta_N,\, i=1,...,\ell\}$ takes place for the first time where $\Gamma_N\cap\Delta_N=\emptyset$ and $q_{i,N},\, i=1,...,\ell$ are certain functions of $n$ taking on nonnegative integer values when $n$ runs from 0 to $N$. The dependence of $q_{i,N}(n)$'s on both $n$ and $N$ is the main novelty of the paper. Under some restrictions on the functions $q_{i,N}$ we obtain Poisson distributions limits of $\mathcal{N}_N$ when counting is until $N$ as $N\to\infty$ and geometric distributions limits when counting is until $\tau_N$ as $N\to\infty$. We obtain also similar results in the dynamical systems setup considering a $\psi$-mixing shift $T$ on a sequence space $\Omega$ and studying the number of multiple returns $\{ T^{q_{i,N}(n)}\omega\in A^a_n,\, i=1,...,\ell\}$ until the first occurrence of another multiple return $\{ T^{q_{i,N}(n)}\omega\in A^b_m,\, i=1,...,\ell\}$ where $A^a_n,\, A_m^b$ are cylinder sets of length $n$ and $m$ constructed by sequences $a,b\in\Omega$, respectively, and chosen so that their probabilities have the same order.