Large deviations for the largest eigenvalue of generalized sample covariance matrices

Abstract: We establish a large-deviations principle for the largest eigenvalue of a generalized sample covariance matrix, meaning a matrix proportional to $Z^T \Gamma Z$, where $Z$ has i.i.d. real or complex entries and $\Gamma$ is not necessarily the identity. We treat the classical case when $Z$ is Gaussian and $\Gamma$ is positive definite, but we also cover two orthogonal extensions: Either the entries of $Z$ can instead be sharp sub-Gaussian, a class including Rademacher and uniform distributions, where we find the same rate function as for the Gaussian model; or $\Gamma$ can have negative eigenvalues if $Z$ remains Gaussian. The latter case confirms formulas of Maillard in the physics literature. We also apply our techniques to the largest eigenvalue of a deformed Wigner matrix, real or complex, where we upgrade previous large-deviations estimates to a full large-deviations principle. Finally, we remove several technical assumptions present in previous related works.