Convergence of the largest eigenvalue of normalized sample covariance matrices when \(p\) and \(n\) both tend to infinity with their ratio converging to zero (Q1932236)

The authors study the distributional properties of the largest eigenvalue of the properly normalized sample covariance matrix constructed from the data matrix whose entries are independent and identically distributed with zero means and unit variances as well as with bounded fourth moments. It is proved that the largest eigenvalue tends almost surely to one. Moreover, the probability of its deviation from one is investigated. The application to the estimation of the population covariance matrix, which is not diagonal, is provided by bounding the spectral norm of the difference between the sample covariance matrix and the corresponding population one.