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On eigenvalues of a renormalized sample correlation matrix

Qianqian Jiang, Junpeng Zhu, Zeng Li

TL;DR

This work analyzes the eigenvalues of a renormalized sample correlation matrix $\mathbf{B}_n$ in ultrahigh-dimensional settings where $p/n\to c\in(0,\infty]$. It develops a unified spectral theory that covers both the Marčenko–Pastur regime and the ultrahigh regime, deriving the limiting spectral distribution (LSD), the limit of the largest eigenvalue, and a central limit theorem (CLT) for linear spectral statistics (LSS) of $\mathbf{B}_n$, with explicit formulas in terms of the Stieltjes transform $s_c(z)$ and, when $c=\infty$, the semicircular law. The theory is then applied to an independence test using $T=\operatorname{tr}\mathbf{B}_n^2$, whose null distribution is asymptotically standard normal, providing a practical diagnostic for both high and ultrahigh dimensions. Simulations confirm the accuracy of the LSD, CLT, and test, demonstrating robustness to Gaussian and non-Gaussian data and a wide range of $(p,n)$-scaling. The results offer a cohesive framework for spectral analysis of renormalized correlation matrices across dimensional regimes and furnish a scalable tool for high-dimensional independence testing.

Abstract

This paper studies the asymptotic spectral properties of a renormalized sample correlation matrix, including the limiting spectral distribution, the properties of largest eigenvalues, and the central limit theorem for linear spectral statistics. All asymptotic results are derived under a unified framework where the dimension-to-sample size ratio $p/n\rightarrow c\in (0,\infty]$. Based on our CLT result, we propose an independence test statistic capable of operating effectively in both high and ultrahigh dimensional scenarios. Simulation experiments demonstrate the accuracy of theoretical results.

On eigenvalues of a renormalized sample correlation matrix

TL;DR

This work analyzes the eigenvalues of a renormalized sample correlation matrix in ultrahigh-dimensional settings where . It develops a unified spectral theory that covers both the Marčenko–Pastur regime and the ultrahigh regime, deriving the limiting spectral distribution (LSD), the limit of the largest eigenvalue, and a central limit theorem (CLT) for linear spectral statistics (LSS) of , with explicit formulas in terms of the Stieltjes transform and, when , the semicircular law. The theory is then applied to an independence test using , whose null distribution is asymptotically standard normal, providing a practical diagnostic for both high and ultrahigh dimensions. Simulations confirm the accuracy of the LSD, CLT, and test, demonstrating robustness to Gaussian and non-Gaussian data and a wide range of -scaling. The results offer a cohesive framework for spectral analysis of renormalized correlation matrices across dimensional regimes and furnish a scalable tool for high-dimensional independence testing.

Abstract

This paper studies the asymptotic spectral properties of a renormalized sample correlation matrix, including the limiting spectral distribution, the properties of largest eigenvalues, and the central limit theorem for linear spectral statistics. All asymptotic results are derived under a unified framework where the dimension-to-sample size ratio . Based on our CLT result, we propose an independence test statistic capable of operating effectively in both high and ultrahigh dimensional scenarios. Simulation experiments demonstrate the accuracy of theoretical results.
Paper Structure (18 sections, 12 theorems, 113 equations, 1 figure, 2 tables)

This paper contains 18 sections, 12 theorems, 113 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Main Results
  3. Preliminaries
  4. LSD of $\mathbf{B}_n$
  5. The largest eigenvalue of $\mathbf{B}_n$
  6. CLT for LSS of $\mathbf{B}_n$
  7. Application of CLTs to hypothesis test
  8. Simulations
  9. Limiting spectral distribution
  10. CLT for LSS
  11. Hypothesis test
  12. Proofs
  13. Notations
  14. Proof of Theorem \ref{['ult-thm:LSD']}
  15. Proof of Theorem \ref{['cor-extreigen']}
  16. ...and 3 more sections

Key Result

Theorem 2.4

Under Assumptions asp1 - asp3, with probability one, the ESD of $\mathbf{B}_{n}$ converges weakly to a (non-random) probability measure $F^c(x)$, which has a density function and has a point mass $1-c$ at the point $-\sqrt{c}$ if $0<c\leq 1$. The Stieltjes transform of $F^c(x)$ is Moreover, the expression of the moments are where $\beta_0=1$ and $\beta_j=\sum_{r=0}^{j-1} \frac{1}{r+1}\binom{j}{

Figures (1)

  • Figure 1: Histograms of sample eigenvalues of $\mathbf{B}_{n}$, fitted by LSD (blue solid curves). In the first row, $(p,n)=(10^4,5000)$, in the second row, $(p,n)=(200^2,200)$, in the third row $(p,n)=(200^{2.5},200)$.

Theorems & Definitions (17)

  • Theorem 2.4
  • Remark 1
  • Theorem 2.5
  • Remark 2
  • Theorem 2.6
  • Remark 3
  • Theorem 2.7
  • Remark 4
  • Theorem 2.8
  • Remark 5
  • ...and 7 more