Let $(X_{nj};1\le j\le m_n,n\ge 1)$ be an array of rowwise ${\mathbb{R}}^d$-valued martingale difference ($d\ge 1$) with respect to σ-fields $({\mathcal{F}}_{nj};0\le j\le m_n,n\ge 1)$ and let $(C_{nj};1\le j\le m_n,n\ge 1)$ be an array of $m\times d$ matrices of real numbers, where $(m_n;n\ge 1)$ is a sequence of positive integers such that $m_n\to \mathrm{\infty }$ as $n\to \mathrm{\infty }$. The aim of this paper is to establish convergence in mean and central limit theorems for weighted sums type $S_n=\sum _{j=1}^{m_n}{C}_{nj}{X}_{nj}$ 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.

让 $(X_{nj};1\le j\le {米}_n,n\ge 1)$ 是一个 rowwise 数组 ${\mathbb{电阻}}^d$- 定值鞅差 ($d\ge 1$) 关于σ场$({\mathcal{F}}_{nj};0\le j\le {米}_n,n\ge 1)$ 然后让 $(C_{nj};1\le j\le {米}_n,n\ge 1)$ 是一个数组 $米\times d$ 实数矩阵，其中 $({米}_n;n\ge 1)$ 是一个正整数序列，使得 ${米}_n\to \mathrm{\infty }$ 作为 $n\to \mathrm{\infty }$. 本文的目的是建立加权和类型的均值和中心极限定理的收敛性${秒}_n=\sum _{j=1}^{{米}_n}{C}_{nj}{X}_{nj}$在无穷远处缓慢变化的某些条件下。我们还应用获得的结果来研究一些统计模型中估计的渐近特性。此外，还给出了两个说明性示例及其模拟。这项研究的动机是在经济学、电信、水文学和物理学应用中出现的模型，这些应用中的创新通常相互依赖并具有无限差异。