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.
Categories: math.PR