Asymptotic Freeness for Rectangular Random Matrices and Large Deviations for Sample Covariance Matrices With Sub-Gaussian Tails

TL;DR

This paper proves a large deviation principle for the spectral distribution of sample covariance matrices with sub-Gaussian entries, extending previous results for Wigner matrices, and introduces an asymptotic freeness result for rectangular free convolution.

Contribution

It establishes a large deviation principle for sub-Gaussian sample covariance matrices and develops an asymptotic freeness result for rectangular free convolution.

Findings

Large deviation principle for empirical spectral measure

Asymptotic freeness result for rectangular free convolution

Bound in the subordination formula for information-plus-noise matrices

Abstract

We establish a large deviation principle for the empirical spectral measure of a sample covariance matrix with sub-Gaussian entries, which extends Bordenave and Caputo's result for Wigner matrices having the same type of entries [7]. To this aim, we need to establish an asymptotic freeness result for rectangular free convolution, more precisely, we give a bound in the subordination formula for information-plus-noise matrices.