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, for all $\varepsilon > 0$, $\sum_{n=1}^{\infty} c_{n} \mathbb{E} \bigg[{\displaystyle \max_{1 \leqslant i \leqslant k_{n}}} \Big\lvert\sum_{j=1}^{i} (X_{n,j} - \mathbb{E} \, X_{n,j} I_{\left\{\lvert X_{n,j} \rvert \leqslant δ\right\}}) \Big\rvert - \varepsilon \bigg]_{+}^{p} < \infty,$ where $x_{+}$ denotes the positive part of $x$ and $p \geqslant 1$, $δ> 0$. Our statements are announced in a general setting allowing to conclude the previous convergence for well-known dependent structures. As an application, we study complete consistency and consistency in the $r$th mean of cumulative sum type estimators of the change in the mean of dependent observations.