A method for estimating the power of moments.

Let X be an observable random variable with unknown distribution function [Formula: see text], [Formula: see text], and let [Formula: see text] We call θ the power of moments of the random variable X. Let [Formula: see text] be a random sample of size n drawn from [Formula: see text]. In this paper we propose the following simple point estimator of θ and investigate its asymptotic properties: [Formula: see text] where [Formula: see text], [Formula: see text]. In particular, we show that [Formula: see text] This means that, under very reasonable conditions on [Formula: see text], [Formula: see text] is actually a consistent estimator of θ.

Some mean convergence theorems for arrays of rowwise pairwise negative quadrant dependent random variables.

For arrays of rowwise pairwise negative quadrant dependent random variables, conditions are provided under which weighted averages converge in mean to 0 thereby extending a result of Chandra, and conditions are also provided under which normed and centered row sums converge in mean to 0. These results are new even if the random variables in each row of the array are independent. Examples are provided showing (i) that the results can fail if the rowwise pairwise negative quadrant dependent hypotheses are dispensed with, and (ii) that almost sure convergence does not necessarily hold.