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样本分块相关矩阵的谱统计量AI 解读

A fundamental concept in multivariate statistics, the sample correlation matrix, is often used to infer the correlation/dependence structure among random variables, when the population mean and covariance are unknown. A natural block extension of it, the sample block correlation matrix, is proposed to take on the same role, when random variables are generalized to random subvectors. In this paper, we establish a spectral theory of the sample block correlation matrices and…

含发散尖峰的大维样本协方差矩阵线性谱统计量的中心极限定理AI 解读

In this paper, we establish the central limit theorem (CLT) for linear spectral statistics (LSSs) of a large-dimensional sample covariance matrix when the population covariance matrices are involved with diverging spikes. This constitutes a nontrivial extension of the Bai–Silverstein theorem (BST) (Ann. Probab. 32 (2004) 553–605), a theorem that has strongly influenced the development of high-dimensional statistics, especially in the applications of random matrix theory to…

高维高斯向量的典型相关系数：有限秩情形AI 解读

Consider a Gaussian vector $\mathbf{z}=(\mathbf{x}',\mathbf{y}')'$, consisting of two sub-vectors $\mathbf{x}$ and $\mathbf{y}$ with dimensions $p$ and $q$, respectively. With $n$ independent observations of $\mathbf{z}$, we study the correlation between $\mathbf{x}$ and $\mathbf{y}$, from the perspective of the canonical correlation analysis. We investigate the high-dimensional case: both $p$ and $q$ are proportional to the sample size $n$. Denote by $\Sigma_{uv}$ the…

椭圆分布下的高维协方差矩阵及其球形性检验应用AI 解读

This paper discusses fluctuations of linear spectral statistics of high-dimensional sample covariance matrices when the underlying population follows an elliptical distribution. Such population often possesses high order correlations among their coordinates, which have great impact on the asymptotic behaviors of linear spectral statistics. Taking such kind of dependency into consideration, we establish a new central limit theorem for the linear spectral statistics in this…