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 $T$ and form a sample correlation matrix $\mathbf{C}$. 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 $\mathbf{C}$ almost surely converges weakly to Mar\v{c}enko-Pastur distribution such that the index is $Q$ and the scale parameter is the limiting ratio of the specific variance to the $i$-th variable $(i\to\infty)$. For an $N$-dimensional normal population with equi-correlation coefficient $\rho$, which is a one-factor model, for the largest eigenvalue $\lambda$ of $\mathbf{C}$, we prove that $\lambda/N$ converges to the equi-correlation coefficient $\rho$ almost surely. These results suggest an important role of an equi-correlated normal population and a factor model in (Laloux et al. Random matrix theory and financial correlations, Int. J. Theor. Appl. Finance, 2000): the histogram of the eigenvalue of sample correlation matrix of the returns of stock prices fits the density of Mar\v{c}enko-Pastur distribution of index $T/N $ and scale parameter $1-\lambda/N$. 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 $N$.