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.

翻译：我们研究了由多水平方差分量产生的样本协方差矩阵。具体而言，令 $ B_n=\frac{1}{N}\sum_{j=1}^NT_{j}^{1/2}x_jx_j^TT_{j}^{1/2}$，其中 $x_j\in R^n$ 为独立同分布的标准高斯变量，$T_{j}=\sum_{r=1}^kl_{jr}^2\Sigma_{r}$ 是 $n\times n$ 实对称矩阵，其谱范数有界，对应于 $k$ 个水平的变异。当矩阵维数 $n$ 和 $N$ 成比例增长时，我们证明了 $B_n$ 的线性谱统计量（LSS）具有高斯极限。该中心极限定理表述为一组 LSS 在通过依赖于 $n$ 和 $N$ 且可数值计算的均值向量 $\Gamma_n$ 和协方差矩阵 $\Lambda_n$ 中心化后，收敛于一个标准多元高斯分布。我们的研究动机源于数量遗传学中表型性状间高维协方差矩阵的估计问题，特别是在具有多达 $k$ 层随机性的嵌套线性随机效应模型内。我们的证明建立在 Bai-Silverstein \cite{baisilverstein2004} 鞅方法的基础上，并引入了一些创新以处理多水平设定。