Correlation Matrix Of Equi-Correlated Normal Population: Fluctuation Of The Largest Eigenvalue, Scaling Of The Bulk Eigenvalues, And Stock Market

Given an N-dimensional sample of size T, form a sample correlation matrix C. Suppose that N and T tend to infinity with T/N converging to a fixed finite constant Q>0. If the population is a factor model, then the eigenvalue distribution of C almost surely converges weakly to MarÄ enko–Pastur distribution such that the index is Q and the scale parameter is the limiting ratio of the specific variance to the ith variable (i→∞). For an N-dimensional normal population with equi-correlation coefficient Ï , which is a one-factor model, for the largest eigenvalue λ of C, we prove that λ/N converges to the equi-correlation coefficient Ï almost surely. These results suggest an important role of an equi-correlated normal population and a factor model in Laloux et al. [(2000) Random matrix theory and financial correlations, International Journal of Theoretical and Applied Finance3 (3), 391–397]: the histogram of the eigenvalue of sample correlation matrix of the returns of stock prices fits the density of MarÄ enko–Pastur distribution of index T/N and scale parameter 1−λ/N. Moreover, we provide the limiting distribution of the largest eigenvalue of a sample covariance matrix of an equi-correlated normal population. We discuss the phase transition as to the decay rate of the equi-correlation coefficient in N.