We investigate the asymptotic normality of distributions of the sequence ∑k∈ℤun,k⁢Xk{\sum_{k\in\mathbb{Z}}u_{n,k}X_{k}}, n∈ℕ{n\in\mathbb{N}}, where (Xk,k∈ℤ){(X_{k},k\in\mathbb{Z})} either is a sequence of i.i.d. random elements or constitutes a linear process with i.i.d. innovations in a separable Hilbert space. The weights (un,k){(u_{n,k})} are in general a family of linear bounded operators. This model includes operator weighted sums of Hilbert space valued linear processes, operator-wise discounted sums in a Hilbert space as well some extensions of classical summation methods.

中文翻译：

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希尔伯特空间值随机元素之和的渐近正态性

我们研究了序列 ∑k∈ℤun,k⁢Xk{\sum_{k\in\mathbb{Z}}u_{n,k}X_{k}}, n∈ℕ{n\in \mathbb{N}}，其中 (Xk,k∈ℤ){(X_{k},k\in\mathbb{Z})} 要么是一个 iid 随机元素序列，要么构成一个具有 iid 创新的线性过程可分希尔伯特空间。权重 (un,k){(u_{n,k})} 通常是一系列线性有界算子。该模型包括希尔伯特空间值线性过程的算子加权和、希尔伯特空间中的算子贴现和以及经典求和方法的一些扩展。