Extreme eigenvalues of sample covariance and correlation matrices
Research output: Book/Report › Ph.D. thesis › Research
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
< 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 highdimensional
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.
Original language | English |
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Publisher | Department of Mathematical Sciences, Faculty of Science, University of Copenhagen |
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Publication status | Published - 2017 |
ID: 174209118