Sankhya: The Indian Journal of Statistics

2004, Volume 66, Pt. 1, 35--48

The Limiting Distributions of Eigenvalues of Sample Correlation Matrices

By

Tiefeng Jiang, University of Minnesota, Minneapolis, USA

SUMMARY. 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. %Rewrite $X=(x_1, x_2, \cdots, x_p).$ Let $R_n=(\rho_{ij})$ be the $p\times p$ 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 Mar\v{c}enko-Pastur law and the semi-circular law, respectively. This work is motivated by testing the hypothesis, assuming population distribution $N_p(\mu, \Sigma),$ that the $p$ variates are uncorrelated.

AMS (1991) subject classification}. 60F05, 60F15, 62H20.

Key words and phrases. Sample correlation coefficient matrix, largest eigenvalue, empirical distributions of eigenvalues, semi-circular law,  Mar\v{c}enko-Pastur law.

Full paper (PDF)