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 Γ1, Γ2, … we show that with probability one (formula presented) where qi(n), i=1, …, ℓ are integer valued functions satisfying certain assumptions and IΓ 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 Γqi(n) is replaced by T−qi(n)C(i)n where C(i)n,i=1,…,ℓ,n≥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.