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 $\sum_{n=1}^{\infty} c_{n} \mathbb{E} \left[\abs{\sum_{k=1}^{n} (X_{n,k} - \mathbb{E} \, X_{n,k})}/b_{n} - \varepsilon \right]_{+}^{p} < \infty$, 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.