Eigenvalue distribution of large sample covariance matrices of linear processes

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 i = 1, . . . , p is modelled as a linear process (Xi;t)t=1...,n = (Σ∞j=0 cjZi;t−j )t=1,...,n, where {Zi;t} are assumed to be independent random variables with finite fourth moments. If the sample size n and the number of variables p = pn both converge to infinity such that y = limn→∞ n/pn > 0, then the empirical spectral distribution of p−1XXT converges to a non-random distribution which only depends on y and the spectral density of (X1;t)t∈Z. In particular, our results apply to (fractionally integrated) ARMA processes, which will be illustrated by some examples.