On the rates of convergence for sums of dependent random variables

For a sequence $\{X_n,n⩾1\}$ of nonnegative random variables where $\max [\min (X_n-s,t),0]$, $t>s⩾0$, satisfy a moment inequality, sufficient conditions are given under which $\sum_{k=1}^n (X_k - $\mathbb{E}$ , X_k)/b_n \overset{$\mathrm{a}\mathrm{.}\mathrm{s}\mathrm{.}$}{\longrightarrow} 0$. Our statement allows us to obtain a strong law of large numbers for sequences of pairwise negatively quadrant dependent random variables under sharp normalising constants.