We establish local laws for sample covariance matrices $K = N^{-1}\sum_{i=1}^N \g_i\g_i^*$ where the random vectors $\g_1, \ldots, \g_N \in \R^n$ are independent with common covariance $Σ$. Previous work has largely focused on the separable model $\g = Σ^{1/2}\w$ with $\w$ having independent entries, but this structure is rarely present in statistical applications involving dependent or nonlinearly transformed data. Under a concentration assumption for quadratic forms $\g^*A\g$, we prove an optimal averaged local law showing that the Stieltjes transform of $K$ converges to its deterministic limit uniformly down to the optimal scale $η\geq N^{-1+\eps}$. Under an additional structural assumption on the cumulant tensors of $\g$ -- which interpolates between the highly structured case of independent entries and generic dependence -- we establish the full anisotropic local law, providing entrywise control of the resolvent $(K-zI)^{-1}$ in arbitrary directions. We discuss several classes of non-separable examples satisfying our assumptions, including conditionally mean-zero distributions, the random features model $\g = σ(X\w)$ arising in machine learning, and Gaussian measures with nonlinear tilting. The proofs introduce a tensor network framework for analyzing fluctuation averaging in the presence of higher-order cumulant structure.

翻译：本文针对样本协方差矩阵 $K = N^{-1}\sum_{i=1}^N \g_i\g_i^*$ 建立了局部定律，其中随机向量 $\g_1, \ldots, \g_N \in \R^n$ 相互独立且具有公共协方差 $Σ$。先前研究主要集中于可分离模型 $\g = Σ^{1/2}\w$（其中 $\w$ 具有独立分量），但该结构在涉及依赖数据或非线性变换数据的统计应用中极少出现。在二次型 $\g^*A\g$ 满足集中性假设的条件下，我们证明了最优平均局部定律，表明 $K$ 的 Stieltjes 变换在最优尺度 $η\geq N^{-1+\eps}$ 范围内一致收敛于其确定性极限。在对 $\g$ 的累积量张量附加结构假设（该假设在高度结构化的独立分量情形与一般依赖情形之间进行插值）下，我们建立了完整的各向异性局部定律，为预解式 $(K-zI)^{-1}$ 在任意方向上的分量提供了逐项控制。我们讨论了满足假设的几类非可分离示例，包括条件均值零分布、机器学习中出现的随机特征模型 $\g = σ(X\w)$，以及具有非线性倾斜的高斯测度。证明过程引入了张量网络框架，用于分析高阶累积量结构下的涨落平均现象。