Resolvent convergence for sample second-moment matrices with heterogeneous profiles under quadratic-form control

Abstract

We study the resolvent \(G^z=\left(\frac{1}{n}XX^{\top}-zI_p\right)^{-1}\), where \(z\in\mathbb{C}\) satisfies \(\Im(z)>0\) and \(X=(x_1,\ldots,x_n)\in\mathbb{R}^{p\times n}\) is a random matrix with independent, but not necessarily identically distributed, columns. The columns are real-valued and have finite second moments, but need not be centered. We identify a deterministic equivalent \(\tilde G^z\) through a finite-dimensional fixed-point system depending on the full second-moment profile \((\mathbb{E}[x_ix_i^{\top}])_{i\in[n]}\). Our quantitative, dimension-dependent bounds are expressed in terms of moments of the centered quadratic forms \(q_i(A):=x_i^{\top}Ax_i-\mathbb{E}[x_i^{\top}Ax_i]\), normalized either by the Hilbert--Schmidt norm or by the operator norm of \(A\). In particular, no independence between the entries of a given column is required. We prove quantitative comparison bounds between \(\operatorname{tr}(BG^z)\) and \(\operatorname{tr}(B\tilde G^z)\) in several regimes. We first consider heterogeneous profiles with uniformly bounded operator norm and bounded aspect ratio. Sharper, profile-adapted arguments then cover uniformly bounded Hilbert--Schmidt second moments without any aspect-ratio condition, a common profile for which additional operator-norm estimates are obtained, and profiles taking \(k\) pairwise commuting values, for which the Hilbert--Schmidt bounds incur a linear loss in \(k\). All probabilistic estimates are global for a fixed \(z\in\mathbb{H}\); the regime \(\Im(z)\downarrow0\) is not considered. The Hilbert--Schmidt estimate yields an explicit random-to-deterministic convergence rate and recovers the Marchenko--Pastur limit in the centered i.i.d. setting under the existence of a moment strictly larger than two.

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