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Spectral analysis of spatial-sign covariance matrices for heavy-tailed data with dependence

Hantao Chen, Cheng Wang

TL;DR

The paper establishes the LSD and a central limit theorem for the spectral statistics of the self-normalized spatial-sign covariance matrix under α-regularly varying heavy-tailed data with general covariance Σ. By leveraging concentration properties of self-normalized random variables and precise quadratic-form analyses, it proves that the MP equation governs the LSD under α≥2 and that a CLT for linear spectral statistics holds under α>4, with explicit mean and covariance structures that depend on the tail index through τ=E[Z^4]. The results extend classical random-matrix theory to robust, self-normalized constructions in heavy-tailed, dependent settings, and are shown to be nearly optimal in tail requirements. These findings provide principled, robust spectral tools for high-dimensional multivariate analysis when data exhibit heavy tails and dependence, with practical implications for robust PCA and related methods.

Abstract

This paper investigates the spectral properties of spatial-sign covariance matrices, a self-normalized version of sample covariance matrices, for data from $α$-regularly varying populations with general covariance structures. By exploiting the elegant properties of self-normalized random variables, we establish the limiting spectral distribution and a central limit theorem for linear spectral statistics. We demonstrate that the Mar{uc}enko-Pastur equation holds under the condition $α\geq 2$, while the central limit theorem for linear spectral statistics is valid for $α>4$, which are shown to be nearly the weakest possible conditions for spatial-sign covariance matrices from heavy-tailed data in the presence of dependence.

Spectral analysis of spatial-sign covariance matrices for heavy-tailed data with dependence

TL;DR

The paper establishes the LSD and a central limit theorem for the spectral statistics of the self-normalized spatial-sign covariance matrix under α-regularly varying heavy-tailed data with general covariance Σ. By leveraging concentration properties of self-normalized random variables and precise quadratic-form analyses, it proves that the MP equation governs the LSD under α≥2 and that a CLT for linear spectral statistics holds under α>4, with explicit mean and covariance structures that depend on the tail index through τ=E[Z^4]. The results extend classical random-matrix theory to robust, self-normalized constructions in heavy-tailed, dependent settings, and are shown to be nearly optimal in tail requirements. These findings provide principled, robust spectral tools for high-dimensional multivariate analysis when data exhibit heavy tails and dependence, with practical implications for robust PCA and related methods.

Abstract

This paper investigates the spectral properties of spatial-sign covariance matrices, a self-normalized version of sample covariance matrices, for data from -regularly varying populations with general covariance structures. By exploiting the elegant properties of self-normalized random variables, we establish the limiting spectral distribution and a central limit theorem for linear spectral statistics. We demonstrate that the Mar{uc}enko-Pastur equation holds under the condition , while the central limit theorem for linear spectral statistics is valid for , which are shown to be nearly the weakest possible conditions for spatial-sign covariance matrices from heavy-tailed data in the presence of dependence.

Paper Structure

This paper contains 26 sections, 19 theorems, 201 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Heavy-tailed populations
  4. Self-normalized random variables
  5. Quadratic forms
  6. Main results
  7. Limiting spectral distribution
  8. Central limit theorem
  9. Numerical results
  10. Empirical performance of spectral distribution
  11. Finite sample performance of CLT
  12. Auxiliary lemmas
  13. Proofs of Self-normalization Propositions
  14. Proof of Proposition \ref{['lem:even_number']}
  15. Proof of Proposition \ref{['lem:odd_number_2']}
  16. ...and 11 more sections

Key Result

Proposition 2.1

If $Z_1,Z_2,\cdots$ are i.i.d. regularly varying with $\alpha\geq 2$, for fixed positive integer $r>0$, we have

Figures (3)

  • Figure 1: The ESDs of ${\bf B}$ with different values of varying index $\alpha\in\{0.5,1,2,4\}$ where $(n,p)=(400,200)$ and the Marc̆enko-Pastur law with $y=0.5$.
  • Figure 2: The infinite norm between ESDs and the generalized Marc̆enko-Pastur law with the growth of $\alpha$ in different settings of dimensions $(n,p)$.
  • Figure 3: The histograms of statistics $\hbox{tr}({\bf B}^2)$ and the theoretical CLT for $\alpha=4.5$ and $n=p$.

Theorems & Definitions (35)

  • Definition 2.1: $\alpha$-regularly varying distribution
  • Example : Student's $t$-distribution
  • Example : Pareto distribution
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Proposition 2.6
  • Proposition 2.7
  • ...and 25 more