Convergence in mean and central limit theorems for weighted sums of martingale difference random vectors with infinite rth moments

Abstract

Let be an array of rowwise -valued martingale difference () with respect to σ-fields and let be an array of matrices of real numbers, where is a sequence of positive integers such that as . The aim of this paper is to establish convergence in mean and central limit theorems for weighted sums type under some conditions of slow variation at infinity. We also apply the obtained results to study the asymptotic properties of estimates in some statistical models. In addition, two illustrative examples and their simulation are given. This study is motivated by models arising in economics, telecommunications, hydrology, and physics applications where the innovations are often dependent on each other and have infinite variances.