We compute the limiting distributions of the largest eigenvalue of a complex Gaussian sample covariance matrix when both the number of samples and the number of variables in each sample become large.

Tiefeng Jiang, The Limiting Distributions of Eigenvalues of Sample Correlation Matrices, Sankhyā: The Indian Journal of Statistics (2003-2007), Vol. 66, No. 1 (Feb., 2004), pp. 35-48 ...