Weighted sums of associated variables

We study the convergence of weighted sums of associated random variables. The convergence for the typical n1/pn1/p normalization is proved assuming finiteness of moments somewhat larger than pp, but still smaller than 2, together with suitable control on the covariance structure described by a truncation that generates covariances that do not grow too quickly. We also consider normalizations of the form n1/qlog1/γnn1/qlog1/γn, where qq is now linked with the properties of the weighting sequence. We prove the convergence under a moment assumption than is weaker that the usual existence of the moment-generating function. Our results extend analogous characterizations known for sums of independent or negatively dependent random variables.