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 ${\Sigma }_i=\mathbb{E}[x_i{x}_i^{\top }]$. For deterministic matrices \(A\), we prove finite-dimensional \(L^q\) bounds for $\tr(A(G^z-\tilde G^z))$ 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 $\sqrt p\sup_{i\in[n]}|Σ_i|_{\hs}=O(n)$, the Hilbert-Schmidt bounds involve approximation of the profile by commuting positive-semidefinite matrices, with explicit dependence on the approximation error and the dimension of their span. We also obtain corresponding bounds for \(\tr(A G^z B G^z)\), with deterministic \(A,B\). All bounds are locally uniform in $z\in \mathbb{C}\setminus [0,\infty )$.

arXiv commentMain text 62pages, 1 table, 1 figure

References in corpus (4)

• A short proof of the Marchenko-Pastur theorem
• Random Matrix Theory Proves that Deep Learning Representations of GAN-data Behave as Gaussian Mixtures
• Circular law for random matrices with unconditional log-concave distribution
• Concentration of measure and generalized product of random vectors with an application to Hanson-Wright-like inequalities