Eigenanalysis of Large Sample Covariance Matrices

Specialeforsvar ved Bo Zhao

Titel: Eigenanalysis of Large Sample Covariance Matrices

Abstract: This thesis studies the asymptotic properties of the extremal eigenvalues of large-dimensional sample covariance matrices. We consider the case where the dimension of the observations p and the sample size n tend to infinity simultaneously. The largest eigenvalues of the sample covariance matrices are approximated under various dependence structures, growth conditions on p, and moment assumptions of the entry distribution, respectively. We find that the characteristics of the eigenvalues depend significantly on the existence of the fourth moment of the entries and the dependence structure of the matrix. In addition, weak convergence results for point processes of normalised sequences of eigenvalues are derived along with convergence results for the corresponding joint distributions. We introduce a linear factor model with Gaussian noise with the purpose of reducing the dimensions of the data, and estimation methods are considered in this setup. The established results are tested throughout the thesis using Monte Carlo simulations and analyses of S&P 500 stock returns. Especially, the finite time performance of the factor model estimators is assessed in detail through extensive simulations and under changes to the initial model assumptions. 

Vejleder: Thomas Mikosch
Censor: Yuri Goegebeur, SDU