Asymptotic normality for eigenvalue statistics of a general sample covariance matrix when $p/n \to \infty$ and applications

TL;DR

This paper establishes the asymptotic normality of eigenvalue statistics for large sample covariance matrices in ultra-high dimensional settings where the dimension grows faster than the sample size, and applies this to develop new statistical tests.

Contribution

It extends the CLT for eigenvalue statistics to ultra-high dimensional regimes and introduces new tests for covariance structures based on this theory.

Findings

Eigenvalue statistics follow a normal distribution when p/n → ∞.

New tests for covariance structures are effective in ultra-high dimensions.

Simulation confirms finite-sample validity of the asymptotic results.

Abstract

The asymptotic normality for a large family of eigenvalue statistics of a general sample covariance matrix is derived under the ultra-high dimensional setting, that is, when the dimension to sample size ratio $p/n \to \infty$. Based on this CLT result, we first adapt the covariance matrix test problem to the new ultra-high dimensional context. Then as a second application, we develop a new test for the separable covariance structure of a matrix-valued white noise. Simulation experiments are conducted for the investigation of finite-sample properties of the general asymptotic normality of eigenvalue statistics, as well as the second test for separable covariance structure of matrix-valued white noise.