  For an array $\left\{X_{n,j}, \, 1 \leqslant j \leqslant k_{n}, n \geqslant 1
\right\}$ of random variables and a sequence $\{c_{n} \}$ of positive numbers,
sufficient conditions are given under which $\sum_{n=1}^{\infty} c_{n}
\mathbb{E} \bigg[{\displaystyle \max_{1 \leqslant i \leqslant k_{n}}}
\Big\lvert\sum_{j=1}^{i} (X_{n,j} - \mathbb{E} \, X_{n,j} I_{\left\{\lvert
X_{n,j} \rvert \leqslant \delta \right\}}) \Big\rvert - \varepsilon
\bigg]_{+}^{p} < \infty$, where $x_{+}$ denotes the positive part of $x$ and $p
\geqslant 1$, $\delta > 0$. Our statements are announced in a general setting
allowing to conclude the previous convergence for well-known dependent
structures.
