EIGENVALUE DISTRIBUTION OF LARGE SAMPLE COVARIANCE
MATRICES OF LINEAR PROCESSES

Oliver Pfaffel

Eckhard Schlemm

Abstract: We derive the distribution of the eigenvalues of a large sample covariance matrix when
the data is dependent in time. More precisely, the dependence for each variable
is modelled as a linear process , where are
assumed to be independent random variables with finite fourth moments. If the sample
size and the number of variables both converge to infinity such that
, then the empirical spectral distribution of converges to
a non-random distribution which only depends on and the spectral density of .
In particular, our results apply to (fractionally integrated) ARMA processes, which will be
illustrated by some examples.

2000 AMS Mathematics Subject Classification: Primary: 15A52; Secondary:
62M10.

Keywords and phrases: Eigenvalue distribution, fractionally integrated ARMA process,
limiting spectral distribution, linear process, random matrix theory, sample covariance
matrix.