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 Γ₁, Γ₂, … we show that with probability one

$$\displaystyle (\sum _{n=1}^N\prod _{i=1}^\ell P({\Gamma }_{q_i(n)}))^{-1}\sum _{n=1}^N\prod _{i=1}^\ell {\mathbb I}_{{\Gamma }_{q_i(n)}}\to 1\,\,\mbox{as}\,\, N\to \infty $$

where q _i(n), i = 1, …, ℓ are integer valued functions satisfying certain assumptions and ${\mathbb I}_{\Gamma }$ denotes the indicator of Γ. When ℓ = 1 (called the conventional setup) this convergence can be established under ϕ-mixing conditions while when ℓ > 1 (called a nonconventional setup) the stronger ψ-mixing condition is required. These results are extended to shifts T of sequence spaces where ${\Gamma }_{q_i(n)}$ is replaced by $T^{-q_i(n)}C^{(i)}_n$ where $C_n^{(i)},i=1,\ldots ,\ell ,\, n\geq 1$ 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.