一般样本协方差矩阵特征值的几个强收敛定理

IF 0.9 4区数学 Q4 PHYSICS, MATHEMATICAL

Random Matrices-Theory and Applications Pub Date : 2021-11-27 DOI:10.1142/s2010326322500290

Yanqing Yin

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引用次数: 1

摘要

本文的目的是研究样本协方差矩阵在更一般的总体下的谱性质。我们考虑一类形式为[公式:见文]的矩阵，其中[公式:见文]是一个[公式:见文]非随机矩阵，[公式:见文]是一个由i.i.d个标准复数项组成的[公式:见文]矩阵。[公式:见文本]与[公式:见文本]相同，而[公式:见文本]可以任意设置，但不小于[公式:见文本]。我们首先证明了在一些温和的假设下，在概率为1的情况下，对于所有大的[公式:见文]，在所有足够大的[公式:见文]的极限分布支持之外的开放区间所包含的任何闭区间中不存在特征值。然后作为白音定律的推广，得到了特征值极值的强收敛性。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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Some strong convergence theorems for eigenvalues of general sample covariance matrices

The aim of this paper is to investigate the spectral properties of sample covariance matrices under a more general population. We consider a class of matrices of the form [Formula: see text], where [Formula: see text] is a [Formula: see text] nonrandom matrix and [Formula: see text] is an [Formula: see text] matrix consisting of i.i.d standard complex entries. [Formula: see text] as [Formula: see text] while [Formula: see text] can be arbitrary but no smaller than [Formula: see text]. We first prove that under some mild assumptions, with probability 1, for all large [Formula: see text], there will be no eigenvalues in any closed interval contained in an open interval which is outside the supports of the limiting distributions for all sufficiently large [Formula: see text]. Then we get the strong convergence result for the extreme eigenvalues as an extension of Bai-Yin law.

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来源期刊

Random Matrices-Theory and Applications Decision Sciences-Statistics, Probability and Uncertainty

CiteScore

1.90

自引率

11.10%

发文量

期刊介绍： Random Matrix Theory (RMT) has a long and rich history and has, especially in recent years, shown to have important applications in many diverse areas of mathematics, science, and engineering. The scope of RMT and its applications include the areas of classical analysis, probability theory, statistical analysis of big data, as well as connections to graph theory, number theory, representation theory, and many areas of mathematical physics. Applications of Random Matrix Theory continue to present themselves and new applications are welcome in this journal. Some examples are orthogonal polynomial theory, free probability, integrable systems, growth models, wireless communications, signal processing, numerical computing, complex networks, economics, statistical mechanics, and quantum theory. Special issues devoted to single topic of current interest will also be considered and published in this journal.