Correlation matrix of equi-correlated normal population: fluctuation of the largest eigenvalue, scaling of the bulk eigenvalues, and stock market

TL;DR

This paper investigates the eigenvalue behavior of correlation matrices from equi-correlated normal populations, revealing convergence properties, distributional limits, and implications for stock market data analysis.

Contribution

It provides new theoretical results on the asymptotic distribution of eigenvalues and the largest eigenvalue in equi-correlated normal populations, connecting to factor models and financial data.

Findings

Largest eigenvalue scaled by N converges to the correlation coefficient rho

Eigenvalue distribution converges to Marčenko-Pastur law under certain conditions

Limiting distribution of the largest eigenvalue is derived

Abstract

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 $N$.