Resolvent convergence under heterogeneous second-moment profiles and quadratic-form control

Abstract

Let $X=(x_1,\dots ,x_n)\in {\mathbb{R}}^{p\times n}$ have independent real columns with finite second moments, not necessarily centered or identically distributed. For $z\in \mathbb{C}\setminus [0,\infty )$, we compare \[ G^z=\left(\frac1nXX^\top-zI_p\right)^{-1} \] with a deterministic matrix ${\tilde{G}}^z$ defined by the second-moment profile $({\Sigma }_i{)}_{i\in [n]}$, where ${\Sigma }_i=\mathbb{E}[x_i{x}_i^{\top }]$. We prove finite-dimensional $L^q$ bounds for $\text{tr}(A(G^z-{\tilde{G}}^z))$ in terms of moments of centered quadratic forms, without assuming independence among the coordinates of a column. Operator-test bounds impose no trace-growth or aspect-ratio restriction and display explicitly the effect of the sizes and repetitions of the second-moment matrices $({\Sigma }_i{)}_{i\in [n]}$. Under $\sqrt{p}{\sup }_{i\in [n]}|{\Sigma }_i{|}_{\text{HS}}=O(n)$, Hilbert--Schmidt-test bounds allow the heterogeneity of the profile to be controlled through approximation by commuting positive-semidefinite matrices. Additive deformations also yield deterministic equivalents for resolvent sandwiches. All bounds are locally uniform away from $[0,\infty )$.

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