Painleve formulas of the limiting distributions for non-null complex sample covariance matrices

Phase transition of the largest eigenvalue for non-null complex sample covariance matrices

We compute the limiting distributions of the largest eigenvalue of a complex Gaussian sample covariance matrix when both the number of samples and the number of variables in each sample become large. When all but finitely many, say $r$, eigenvalues of the covariance matrix are the same, the dependence of the limiting distribution of the largest eigenvalue of the sample covariance matrix on those distinguished $r$ eigenvalues of the covariance matrix is completely characterized in terms of an infinite sequence of new distribution functions that generalize the Tracy-Widom distributions of the ra…

Tracy-Widom at each edge of real covariance and MANOVA estimators

We study the sample covariance matrix for real-valued data with general population covariance, as well as MANOVA-type covariance estimators in variance components models under null hypotheses of global sphericity. In the limit as matrix dimensions increase proportionally, the asymptotic spectra of such estimators may have multiple disjoint intervals of support, possibly intersecting the negative half line. We show that the distribution of the extremal eigenvalue at each regular edge of the support has a GOE Tracy-Widom limit. Our proof extends a comparison argument of Ji Oon Lee and Kevin Schn…

Asymptotics for the Covariance of the Airy_2 process

In this paper we compute some of the higher order terms in the large-t asymptotic expansion of the Airy process two-point function, extending the previous work of Adler and van Moerbeke and Widom. We prove that it is possible to represent any order asymptotic approximation as a polynomial and integrals of the Hastings-McLeod Painlevé II function and its first derivative. Further, for up to tenth order we give this asymptotic approximation as a linear combination of the Tracy-Widom GUE density function f_2 and its derivatives. As a corollary to this, the asymptotic covariance is expressed up to…

高维椭圆模型中的双样本协方差推断

Two-Sample Covariance Inference in High-Dimensional Elliptical Models

本文针对高维椭圆模型提出一种基于平方 Frobenius 范数 U 统计量的双样本协方差矩阵检验，建立了原假设与备择假设下均成立的中心极限定理，首次为椭圆数据提供理论保证，且无需稀疏性或维数增长条件。

Moments of the complex multivariate normal distribution

We present a characterization of the null moments of the Complex Multivariate Normal Distribution with non-singular covariance matrix and we give closed-forms expressions for its non-null moments.

Spectral analysis of high-dimensional sample covariance matrices with missing observations

We study high-dimensional sample covariance matrices based on independent random vectors with missing coordinates. The presence of missing observations is common in modern applications such as climate studies or gene expression micro-arrays. A weak approximation on the spectral distribution in the "large dimension $d$ and large sample size $n$" asymptotics is derived for possibly different observation probabilities in the coordinates. The spectral distribution turns out to be strongly influenced by the missingness mechanism. In the null case under the missing at random scenario where each comp…