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

Let (X-nj; 1 <= j <= m n ,n >= 1) be an array of rowwise R-d-valued martingale difference (d >= 1) with respect to sigma-fields (F-nj; 0 <= j <= m(n),n >= 1) and let (C-nj; 1 <= j <= m(n), n >= 1) be an array of m x d matrices of real numbers, where (m(n); n >= 1) is a sequence of positive integers such that m(n) -> infinity as n -> infinity. The aim of this paper is to establish convergence in mean and central limit theorems for weighted sums type S-n = Sigma(mn)(j=1) CnjXnj 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.