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

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 ${\mathrm{\Gamma }}_1,{\mathrm{\Gamma }}_2,\mathrm{.}\mathrm{.}\mathrm{.}$ we show that with probability one $${(\sum _{n=1}^N\prod _{i=1}^{\mathrm{\ell }}P({\mathrm{\Gamma }}_{q_i(n)}))}^{-1}\sum _{n=1}^N\prod _{i=1}^{\mathrm{\ell }}{\mathbb{I}}_{{\mathrm{\Gamma }}_{q_i(n)}}\to 1asN\to \mathrm{\infty }$$ where $q_i(n),i=1,\mathrm{.}\mathrm{.}\mathrm{.},\mathrm{\ell }$ are integer valued functions satisfying certain assumptions and ${\mathbb{I}}_{\mathrm{\Gamma }}$ denotes the indicator of $\mathrm{\Gamma }$. When $\mathrm{\ell }=1$ (called the conventional setup) this convergence can be established under $\varphi$-mixing conditions while when $\mathrm{\ell }>1$ (called a nonconventional setup) the stronger $\psi$-mixing condition is required. These results are extended to shifts $T$ of sequence spaces where ${\mathrm{\Gamma }}_{q_i(n)}$ 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.