Almost sure limit of the smallest eigenvalue of sample correlation matrix

Abstract

Suppose we have a data matrix consisting of independent and identically distributed entries with finite fourth moment, we show that the smallest eigenvalue of the sample correlation matrix converges almost surely to a constant provided that the ration of dimensions of the data matrix goes to a positive constant. We accomplish this by establishing a similar result for the sample covariance matrix. The proof relies strongly on existing results about the simplified sample covariance matrix.