  For a sequence $\{X_{n}, \, n \geqslant 1 \}$ of nonnegative random variables
where $\max[\min(X_{n} - s,t),0]$, $t > s \geqslant 0$, satisfy a moment
inequality, sufficient conditions are given under which $\sum_{k=1}^n (X_k -
\mathbb{E} \, X_k)/b_n \overset{\mathrm{a.s.}}{\longrightarrow} 0$. 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.
