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Liberating dimension and spectral norm: A universal approach to spectral properties of sample covariance matrices

Yanqing Yin

TL;DR

This work introduces a universal renormalization of the sample covariance matrix to control spectral properties across ultrahigh-dimensional regimes without requiring restrictive growth rates. It derives the limiting spectral distribution (LSD) of the renormalized matrix and a harmonic central limit theorem (CLT) for linear spectral statistics (LSS) under general population spectra, removing the need for a bounded population spectral norm. The results cover comparable, large-n, and large-p regimes and provide explicit mean and covariance structures for LSS, enabling robust, rate-free inference in high dimensions. As an application, the authors develop Gaussian-limit tests for identity covariance testing (Frobenius-norm and likelihood ratio tests), demonstrating rate-free performance and broader applicability in practical high-dimensional inference.

Abstract

In this paper, our objective is to present a constraining principle governing the spectral properties of the sample covariance matrix. This principle exhibits harmonious behavior across diverse limiting frameworks, eliminating the need for constraints on the rates of dimension $p$ and sample size $n$, as long as they both tend to infinity. We accomplish this by employing a suitable normalization technique on the original sample covariance matrix. Following this, we establish a harmonic central limit theorem for linear spectral statistics within this expansive framework. This achievement effectively eliminates the necessity for a bounded spectral norm on the population covariance matrix and relaxes constraints on the rates of dimension $p$ and sample size $n$, thereby significantly broadening the applicability of these results in the field of high-dimensional statistics. We illustrate the power of the established results by considering the test for covariance structure under high dimensionality, freeing both $p$ and $n$.

Liberating dimension and spectral norm: A universal approach to spectral properties of sample covariance matrices

TL;DR

This work introduces a universal renormalization of the sample covariance matrix to control spectral properties across ultrahigh-dimensional regimes without requiring restrictive growth rates. It derives the limiting spectral distribution (LSD) of the renormalized matrix and a harmonic central limit theorem (CLT) for linear spectral statistics (LSS) under general population spectra, removing the need for a bounded population spectral norm. The results cover comparable, large-n, and large-p regimes and provide explicit mean and covariance structures for LSS, enabling robust, rate-free inference in high dimensions. As an application, the authors develop Gaussian-limit tests for identity covariance testing (Frobenius-norm and likelihood ratio tests), demonstrating rate-free performance and broader applicability in practical high-dimensional inference.

Abstract

In this paper, our objective is to present a constraining principle governing the spectral properties of the sample covariance matrix. This principle exhibits harmonious behavior across diverse limiting frameworks, eliminating the need for constraints on the rates of dimension and sample size , as long as they both tend to infinity. We accomplish this by employing a suitable normalization technique on the original sample covariance matrix. Following this, we establish a harmonic central limit theorem for linear spectral statistics within this expansive framework. This achievement effectively eliminates the necessity for a bounded spectral norm on the population covariance matrix and relaxes constraints on the rates of dimension and sample size , thereby significantly broadening the applicability of these results in the field of high-dimensional statistics. We illustrate the power of the established results by considering the test for covariance structure under high dimensionality, freeing both and .
Paper Structure (28 sections, 11 theorems, 157 equations)

This paper contains 28 sections, 11 theorems, 157 equations.

Table of Contents

  1. Introduction and motivation
  2. Main results
  3. Convergence of the ESD
  4. A harmonic CLT for LSS
  5. Statistical application: Identity covariance matrix test
  6. Frobenius norm test
  7. Likelihood ratio test
  8. Proof of main theorems
  9. Proof of Theorem \ref{['th:1']}
  10. Proof of Theorem \ref{['cltthm']}
  11. Truncation and recentralization
  12. Convergence of Stieltjes transform
  13. The limiting distribution of $M_{n1}(z)$
  14. Convergence of $M_{n2}(z)$
  15. Tightness of $M_{n1}(z)$
  16. ...and 13 more sections

Key Result

Theorem 1

If, as $p\to\infty$ and $n\to\infty$, the following conditions hold: Then, with probability one, the ESD of the renormalized sample covariance matrix $\mathbf S_n$ converges to a probability distribution $F$, whose Stieltjes transform $m(z)$ satisfies and is unique in the set $\left\{m\in\mathbb C^+:-(c_2-c_1)/z+c_1m\in\mathbb C^+\ {\rm or}\ c_2=0\right\}$.

Theorems & Definitions (14)

  • Theorem 1
  • Remark 2
  • Theorem 3
  • Theorem 4
  • Remark 5
  • Theorem 6
  • Remark 7
  • Lemma 8
  • Lemma 9: Corollary A.42 of BaiS10S
  • Lemma 10: Theorem A.44 of BaiS10S
  • ...and 4 more