No Eigenvalues Outside the Limiting Support of Generally Correlated and Noncentral Sample Covariance Matrices

Zeyan Zhuang; Xin Zhang; Dongfang Xu; Shenghui Song

Home/Archive/arXiv

arXiv · 2507.03356

No Eigenvalues Outside the Limiting Support of Generally Correlated and Noncentral Sample Covariance Matrices

Zeyan Zhuang·Xin Zhang·Dongfang Xu·Shenghui Song

View original source ↗Read source document ↗Export metadata ↓

Abstract

Spectral properties of random matrices play an important role in statistics, machine learning, communications, and many other areas. Engaging results regarding the convergence of the empirical spectral distribution (ESD) and the ``no-eigenvalue'' property have been obtained for random matrices with different correlation structures. However, the related spectral analysis for generally correlated and noncentral random matrices is still incomplete, and this paper aims to fill this research gap. Specifically, we consider matrices whose columns are independent but with non-zero means and non-identical correlations. Under high-dimensional asymptotics where both the number of rows and columns grow simultaneously to infinity, we first establish the almost sure convergence of the ESD for the concerned random matrices to a deterministic limit, assuming mild conditions. Furthermore, we prove that with probability 1, no eigenvalues will appear in any closed interval outside the support of the limiting distribution for matrices with sufficiently large dimensions. The above results can be applied to different areas such as statistics, wireless communications, and signal processing. In this paper, we apply the derived results to two communication scenarios: 1) We determine the limiting performance of the signal-to-interference-plus-noise ratio for multi-user multiple-input multiple-output (MIMO) systems with linear minimum mean-square error receivers; and 2) We establish the invertibility of zero-forcing precoding matrices in downlink MIMO systems, providing theoretical guarantees.

Explore related subjects

math.ST stat.TH

Keep this discovery

BibTeX RIS

Zeyan Zhuang, Xin Zhang, Dongfang Xu, Shenghui Song. 2025-07-04. No Eigenvalues Outside the Limiting Support of Generally Correlated and Noncentral Sample Covariance Matrices. https://arxiv.org/abs/2507.03356

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

A Scale Invariance Property of PCA

The PCA algorithm is sensitive to changes in measurement scale. Measuring one variable of a system in inches rather than centimeters, say, alters both its principal axes and principal eigenvalues. Although this scale dependence is generally complicated, we show here that it nevertheless obeys a strict invariance property: under a continuous scale adjustment, the initial state's $k$-th largest principal component (ordered by eigenvalue) continuously evolves into the final state's $k$-th largest principal component, for each $k$. In this sense, we can say that the modes of PCA are "order-stable" with respect to changes in measurement scale. A special case occurs when scaling along directions that are orthogonal to some modes. Here, apparent eigenvalue crossings can occur. However, we show that we can interpret these apparent crossings as cases where the modes instantaneously swap their orientation, in this way maintaining the required order stability.

math.ST↗

arXiv2026

Small noise asymptotics for linear parabolic SPDEs in two space dimensions with unknown damping factors

We study parametric estimation for second order linear parabolic stochastic partial differential equations in two space dimensions with a small volatility parameter driven by a $Q$-Wiener process with an unknown damping parameter using high frequency spatio-temporal data. We first provide an estimator for the damping parameter of the $Q$-Wiener process utilizing realized quadratic variations based on spatial and temporal increments. We next propose minimum contrast estimators of the diffusive and advective parameters in the SPDE using a contrast function with the proposed estimator of the damping parameter. We then construct a quasi-maximum likelihood estimator of the reaction parameter in the SPDE using the approximate coordinate process derived from the estimators of the diffusive and advective parameters. We also provide simulation results of the proposed estimators.

math.ST↗

arXiv2026

Spike Estimation from Heteroscedastic Noise via Random Splitting

In this paper, we consider a spiked Wigner type matrix with a heteroscedastic and unknown variance profile. It is well known that in the supercritical regime of the BBP transition, strong spikes can create outliers in the spectrum. Unfortunately, in the heteroscedastic case, in general it is not possible to estimate the spike strength from these observed outlier consistently, as the latter is a solution to a Dyson equation with unknown parameters from the variance profile. In this paper, inspired by the work on sparse matrix completion \citep{BordenaveCosteNadakuditi2023}, we introduce an asymmetrized model by randomly splitting the spiked matrix into two parts, which transforms the noisy Wigner type matrix into a non Hermitian random matrix, while preserving the Hermitian spikes at the cost of a dilution. We establish a BBP type transition for the asymmetrized model, from which we can estimate the strength of the spikes precisely, even without knowing the variance profile of the noise part. We then further apply our approach to study the correlation between two correlated spiked models, where the spike/signal parts of the two models are correlated, and the noise parts are independent but may both be heteroscedastic. By applying our asymmetrization approach to the two models separately and also jointly, we are able to obtain a precise estimate of the correlation between the signal parts of the two models.

math.ST↗