  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.
