2024/06/06 by Ran Xie, Iain M. Johnstone, Xie, Ran +1
Chemistry · Engineering · Mathematics · #FOS: Mathematics #Fault Detection and Control Systems #Probability (math.PR) #Spectroscopy and Chemometric Analyses #Statistical Methods and Inference #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.2406.03719
openalex publication_date 2024/06/06 · openalex created_date 2024/06/08 · openalex updated_date 2026/07/28
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 \citebaisilverstein2004 martingale method with some innovation to handle the multi-level setting.