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

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 $\text{Tr}(BG^z)$ is close to $\text{Tr}(B\tilde G^z)$, with error controlled by $\|B\|_{\text{HS}}$ under first-moment bounds on the quadratic forms, and by $\|B\|_{\text{HS}}/\sqrt n$ under suitable second-moment bounds.