Resolvent convergence for sample covariance matrices with general covariance profiles and quadratic-form control

TL;DR

This paper investigates the resolvent of sample covariance matrices with general, possibly dependent, columns, providing bounds in terms of quadratic form moments and establishing deterministic equivalents in high-dimensional regimes.

Contribution

It introduces bounds for resolvent convergence that do not require independence within columns and constructs deterministic equivalents based on second moments.

Findings

Trace of the resolvent is close to its deterministic equivalent.

Error bounds depend on the Hilbert-Schmidt norm of the test matrix.

Results hold under minimal moment assumptions on quadratic forms.

Abstract

We study the resolvent \[ G^z = \left(\frac{1}{n}XX^T - zI_p\right)^{-1}, \qquad z\in\mathbb C,\ \Im(z)>0, \] where $X=(x_1,\ldots,x_n)\in\mathcal M_{p,n}$ is a random matrix with independent, but not necessarily identically distributed, columns. Our bounds are expressed in terms of moments of the centered quadratic forms \[ q_i(A):=x_i^TAx_i-\mathbb E[x_i^TAx_i], \] for deterministic matrices $A$ with unit Hilbert--Schmidt norm. In particular, we do not assume independence between the entries of a given column $x_i$. In the quasi-asymptotic regime $p\le O(n)$, the matrix $G^z$ admits a natural deterministic equivalent $\tilde G^z$, depending only on the second moments of the column vectors $x_1,\ldots,x_n$. We show that, for any deterministic matrix $B\in\mathcal M_p$, the trace $\tr(BG^z)$ is close to $\tr(B\tilde G^z)$, with error controlled by $\|B\|_{\hs}$ under first-moment bounds on the quadratic forms, and by $\|B\|_{\hs}/\sqrt n$ under suitable second-moment bounds.