The strong Borel--Cantelli property in conventional and nonconventional setups

Abstract

We study the strong Borel-Cantelli property both for events and for shifts on sequence spaces considering both a conventional and a nonconventional setups. Namely, under certain conditions on events $\Gamma_1,\Gamma_2,...$ we show that with probability one \[ \left(\sum_{n=1}^N\prod_{i=1}^\ell P(\Gamma_{q_i(n)})\right)^{-1}\sum_{n=1}^N\prod_{i=1}^\ell\mathbb{I}_{\Gamma_{q_i(n)}}\to 1\,\,\mbox{as}\,\, N\to\infty \] where $q_i(n),\, i=1,...,\ell$ are integer valued functions satisfying certain assumptions and $\mathbb{I}_\Gamma$ denotes the indicator of $\Gamma$. When $\ell=1$ (called the conventional setup) this convergence can be established under $\phi$-mixing conditions while when $\ell>1$ (called a nonconventional setup) the stronger $\psi$-mixing condition is required. These results are extended to shifts $T$ of sequence spaces where $\Gamma_{q_i(n)}$ is replaced by $T^{-q_i(n)}C_n^{(i)}$ where $C_n^{(i)},\, i=1,...,\ell,\, n\geq 1$ is a sequence of cylinder sets. As an application we study the asymptotical behavior of maximums of certain logarithmic distance functions and of (multiple) hitting times of shrinking cylinders.