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We derive the distribution of the eigenvalues of a large sample covariancematrix when the data is dependent in time. More precisely, the dependence foreach variable $i=1,...,p$ is modelled as a linear process$(X_{i,t})_{t=1,...,n}=(\sum_{j=0}^\infty c_j Z_{i,t-j})_{t=1,...,n}$, where$\{Z_{i,t}\}$ are assumed to be independent random variables with finite fourthmoments. If the sample size $n$ and the number of variables $p=p_n$ bothconverge to infinity such that $y=\lim_{n\to\infty}{n/p_n}>0$, then theempirical spectral distribution of $p^{-1}\X\X^T$ converges to anon\hyp{}random distribution which only depends on $y$ and the spectral densityof $(X_{1,t})_{t\in\Z}$. In particular, our results apply to (fractionallyintegrated) ARMA processes, which we illustrate by some examples.

Eigenvalue distribution of large sample covariance matrices of linear processes