PhD Defense Johannes Heiny

Title: Extreme Eigenvalues of Sample Covariance and Correlation Matrices

Abstract:

This thesis is concerned with asymptotic properties of the eigenvalues of high-dimensional sample covariance and correlation matrices under an infinite fourth moment of the entries.

In the first part, we study the joint distributional convergence of the largest eigenvalues of the sample covariance matrix of a p-dimensional heavy-tailed time series when p converges to infinity together with the sample size n. We generalize the growth rates of p existing in the literature. Assuming a regular variation condition with tail index alpha<4, we employ a large deviations approach to show that the extreme eigenvalues are essentially determined by the extreme order statistics from an array of iid random variables.  The asymptotic behavior of the extreme eigenvalues is then derived routinely from classical extreme value theory. The resulting approximations are strikingly simple considering the high dimension of the problem at hand. We develop a theory for the point process of the normalized eigenvalues of the sample covariance matrix in the case where rows and columns of the data are linearly dependent. Based on the weak convergence of this point process we derive the limit laws of various functionals of the eigenvalues.

In the second part, we show that the largest and smallest eigenvalues of a high-dimensional sample correlation matrix possess almost sure non-random limits if the truncated variance of the entry distribution is ‘’almost slowly varying'', a condition we describe via moment properties of self-normalized sums. We compare the behavior of the eigenvalues of the sample covariance and sample correlation matrices and argue that the latter seems more robust, in particular in the case of infinite fourth moment.

Supervisor: Prof Thomas Mikosch, Math, University of Copenhagen

Co-Supervisor: Olivier Wintenberger, UCPH, IMF and Univ. Paris Marie and Pierre Curie

Assessment committee:

Prof. Jeffrey Collamore (chairman), MATH, University of Copenhagen

Prof. Claudia Klueppelberg, TU Munich

Prof. Richard A. Davis, Columbia University New York