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

We study sample covariance matrices arising from multi-level components of variance. Thus, let $B_n=\frac{1}{N}{\sum }_{j=1}^N{T}_j^{1\mathrm{/}2}{x}_j{x}_j^T{T}_j^{1\mathrm{/}2}$, where $x_j\in R^n$ are i.i.d. standard Gaussian, and $T_j={\sum }_{r=1}^k{l}_{jr}^2{\mathrm{\Sigma }}_r$ 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 ${\mathrm{\Gamma }}_n$ and a covariance matrix ${\mathrm{\Lambda }}_n$ 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.