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

Let X=(x1,ldots,xn)∈mathbb Rp× n have independent real columns with finite second moments, not necessarily centered or identically distributed. For z∈mathbb Csetminus[0,∞), we compare [ Gz=left(frac1nXX^top-zIpright)-1 ] with a deterministic matrix tilde Gz defined by the second-moment profile (Sigmai)i∈[n], where Sigmai=mathbb E[xixi^top]. We prove finite-dimensional Lq bounds for tr(A(Gz-tilde Gz)) 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 (Sigmai)i∈[n]. Under √ psupi∈[n]|Sigmai|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,∞).