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 Bₙ=1/N∑ⱼ₌₁NTⱼ1/2xⱼxⱼTTⱼ1/2, where xⱼ∈ Rⁿ are i.i.d. standard Gaussian, and Tⱼ=∑ᵣ₌₁ᵏlⱼᵣ²Σᵣ are n× 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 Bₙ 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 Γₙ and a covariance matrix Λₙ 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 {baisilverstein2004} martingale method with some innovation to handle the multi-level setting.