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

## TL;DR

This paper investigates the convergence of series involving moments of row sums of random variables in a triangular array, extending results to various dependence structures.

## Contribution

It establishes conditions for the convergence of series of moments of centered row sums under general dependence assumptions.

## Key findings

- Series convergence under various dependence structures.
- Conditions involving moments and sequences for convergence.
- General setting broadening previous results.

## 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.