大维样本协方差阵在高维统计及无线通信中的应用

Nowadays, random matrices find applications in fields as diverse as neural networks, multivariate statistics, information theory, signal processing and so on. This project is mainly to study the general sample covariance matrices which come from communication channels in the communications and information theory. Furthermore, the general sample covariance matrix is the extension of the traditional one. The primary sample matrix consists of independent entries. But now the sample matrix what we concern is obtained by multiplying a nonnegative definite matrix on both sides of the primary one. This leads to the loss of row and column independence...This project aims to establish the central limit theorem of linear spectral statistics of the general sample covariance matrices. And we will proceed with three steps. At first, the two matrices are both nonrandom matrices. Next, either of them is random matrix. Finally, both of them are random. Based on these results, we will deal with the problems in multivariate statistical inference and wireless communication.

随着科学技术的发展，随机矩阵理论的应用越来越广泛，如神经网络、多元统计、信息理论、信号处理等。本项目研究的广义样本协方差矩阵来源于无线通讯中的线性向量无记忆信道模型。另一方面，广义样本协方差矩阵也是对通常意义下样本协方差矩阵的推广，原来的样本一般是由独立变量构成的矩阵，现在研究的样本是在此基础上左右都乘以一个非负定矩阵。这就导致我们所关心的样本行独立性和列独立性都被破坏了。. 本项目旨在确立广义样本协方差阵线性谱统计量的中心极限定理。这些定理为解决无线通讯中信道质量勘测和多元统计中的问题提供了理论支持。由于左右乘的矩阵都是随机的，这给研究工作带来很多困难。为此，我们拟分三步开展研究：.a）左右乘的矩阵均为非随机，.b）随机化其中一个矩阵，另外一个仍为非随机，.c）两个矩阵均随机化。. 最后，探讨如何将研究结果应用到无线通讯和多元统计的实际问题中去。

随着科学技术的发展，随机矩阵理论的应用越来越广泛，如神经网络、多元统计、信息理论、信号处理等。本项目研究的广义样本协方差矩阵，也称为可分样本协方差矩阵。该矩阵模型来源于无线通讯中的线性向量无记忆信道模型。另一方面，可分样本协方差矩阵也是对通常意义下样本协方差矩阵的推广，原来的样本一般是由独立变量构成的矩阵，本项目研究的样本是在此基础上左右都乘以一个非负定矩阵。这就导致我们所关心的样本行独立性和列独立性都被破坏了。.本项目确立了可分样本协方差阵和广义可分样本协方差矩阵线性谱统计量的中心极限定理。这些定理为解决无线通讯中信道质量勘测和多元统计中的某些问题如高维白噪声检验提供了理论支持。