The Limiting Distributions of Eigenvalues of Sample Correlation Matrices

摘要

Let X_n = (x_(ij)) be an n by p data matrix, where the n rows form a random sample of size n from a certain p-dimensional population distribution. Let R_n = (ρ_(ij)) be the p x p sample correlation coefficient matrix of X_n. Assuming that x_(ij)'s are independent and identically distributed (x_(ij)'s are required to be only independent when they are normals), we show that the largest eigenvalue of R_n almost surely converges to a constant provided n/p goes to a positive constant. Under two conditions on the ratio n/p, we show that the empirical distribution of eigenvalues of R_n converges weakly to the Marcenko-Pastur law and the semi-circular law, respectively. This work is motivated by testing the hypothesis, assuming population distribution N_p(μ, Σ), that the p variates are uncorrelated.