On the convergence of series of moments for row sums of random variables

Abstract: Given a triangular array $\left{X_{n,k}, \, 1 \leqslant k \leqslant n, n \geqslant 1 \right}$ of random variables satisfying $\mathbb{E} \lvert X_{n,k} \rvert^{p} < \infty$ for some $p \geqslant 1$ and sequences ${b_{n} }$, ${c_{n} }$ of positive real numbers, we shall prove that $\sum_{n=1}^\infty c_n \mathbb{E} \left[ |\sum_{k=1}^n (X_{n,k} - \mathbb{E} \, X_{n,k})| / b_n - \varepsilon \right]+^p < \infty$, where $x+ = \max(x,0)$. Our results are announced in a general setting, allowing us to obtain the convergence of the series in question under various types of dependence.