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For an array of rowwise pairwise negative quadrant dependent, mean 0 random variables, Chandra, Li, and Rosalsky provided conditions under which weighted averages converge in L1 to 0. The Chandra, Li, and Rosalsky result is extended to Lr convergence (1≤r<2) and is shown to hold under weaker conditions by applying a mean convergence result of Sung and an inequality of Adler, Rosalsky, and Taylor. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Let X be an observable random variable with unknown distribution function F(x)=P(X≤x), −∞, and let θ=sup{r≥0:E|X|r<∞}. We call θ the power of moments of the random variable X. Let X1,X2,…,Xn be a random sample of size n drawn from F(⋅). In this paper we propose the following simple point estimator of θ and investigate its asymptotic properties: θˆn=lognlogmax1≤k≤n|Xk|, where logx=ln(e∨x), −∞. In particular, we show that θˆn→Pθif and only iflimx→∞xrP(|X|>x)=∞∀r>θ. This means that, under very reasonable conditions on F(⋅), θˆn is actually a consistent estimator of θ. [ABSTRACT FROM AUTHOR]