CLT for Linear Spectral Statistics in High-Dimensional Random Effects Models

TL;DR

This paper establishes a central limit theorem for linear spectral statistics of high-dimensional covariance matrices from multi-level random effects models, with applications in quantitative genetics.

Contribution

It extends CLT results for spectral statistics to multi-level variance component models, providing a numerical way to evaluate the Gaussian limits.

Findings

LSS of covariance matrices converge to Gaussian distributions

Mean and covariance depend on model parameters and can be numerically evaluated

Applicable to high-dimensional genetic data analysis

Abstract

We study sample covariance matrices arising from multi-level components of variance. Thus, let $ B_n=\frac{1}{N}\sum_{j=1}^NT_{j}^{1/2}x_jx_j^TT_{j}^{1/2}$, where $x_j\in R^n$ are i.i.d. standard Gaussian, and $T_{j}=\sum_{r=1}^kl_{jr}^2\Sigma_{r}$ are $n\times n$ real symmetric matrices with bounded spectral norm, corresponding to $k$ levels of variation. As the matrix dimensions $n$ and $N$ increase proportionally, we show that the linear spectral statistics (LSS) of $B_n$ have Gaussian limits. The CLT is expressed as the convergence of a set of LSS to a standard multivariate Gaussian after centering by a mean vector $\Gamma_n$ and a covariance matrix $\Lambda_n$ which depend on $n$ and $N$ and may be evaluated numerically. Our work is motivated by the estimation of high-dimensional covariance matrices between phenotypic traits in quantitative genetics, particularly within nested linear random-effects models with up to $k$ levels of randomness. Our proof builds on the Bai-Silverstein \cite{baisilverstein2004} martingale method with some innovation to handle the multi-level setting.