Strong convergence for weighted sums of widely orthant dependent random variables and applications (Q6164842)

The paper develops two general results (Theorem 2.1 and Theorem 2.2) for the complete convergence of weighted sums of widely orthant dependent random variables under a common condition on the strength of the dependence. The sequence of random variables \((X_i)\) is said to be widely orthant dependent if there is a function \(g\) with \[ \mathbb{P}(X_1\leq x_1,\ldots,X_n\leq x_n) \leq g(n) \prod_{i=1}^n \mathbb{P}(X_i\leq x_i) \] and \[ \mathbb{P}(X_1>x_1,\ldots,X_n>x_n) \leq g(n) \prod_{i=1}^n \mathbb{P}(X_i>x_i) \] for all \(n\) and \(x_1,\ldots,x_n\). The two limiting theorems are stated under the assumption that \(g(n)\) grows at no more than a polynomial rate. Applications are given to linear regression, nonparametric regression, and renewal theory. A numerical simulation illustrates some of the findings from these three examples.