A Note on Weighted Sums of Associated Random Variables

We prove the convergence of weighted sums of associated random variables normalized by $n^{1/p}$, $p\in(1,2)$, assuming the existence of moments somewhat larger than $p$, depending on the behaviour of the weights, improving on previous results by getting closer to the moment assumption used for the case of constant weights. Besides moment conditions we assume a convenient behaviour either on truncated covariances or on joint tail probabilities. Our results extend analogous characterizations known for sums of independent or negatively dependent random variables