  This paper is the third chapter of three of the author's undergraduate
thesis. In this paper, we study the convergence of local bulk statistics for
linearized covariance matrices under Dyson's Brownian motion. We consider
deterministic initial data $V$ approximate the Dyson Brownian motion for
linearized covariance matrices by the Wigner flow. Using universality results
for the Wigner flow, we deduce universality for the linearized covariance
matrices. We deduce bulk universality of averaged bulk correlation functions
for both biregular bipartite graphs and honest covariance matrices. We also
deduce a weak level repulsion estimate for the Dyson Brownian motion of
linearized covariance matrices.
