Correlation matrix of equi-correlated normal population: fluctuation of the largest eigenvalue, scaling of the bulk eigenvalues, and stock market
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Given an N-dimensional sample of size T and 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 Marčenko-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→∞). For an N-dimensional normal population with equi-correlation coefficient ρ, which is a one-factor model, for the largest eigenvalue λ of 𝐂, we prove that λ/N converges to the equi-correlation coefficient ρ 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 Marčenko-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.
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