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In this paper we study the joint distributional convergence of the largesteigenvalues of the sample covariance matrix of a $p$-dimensional time serieswith iid entries when $p$ converges to infinity together with the sample size$n$. We consider only heavy-tailed time series in the sense that the entriessatisfy some regular variation condition which ensures that their fourth momentis infinite. In this case, Soshnikov [31, 32] and Auffinger et al. [2] provedthe weak convergence of the point processes of the normalized eigenvalues ofthe sample covariance matrix towards an inhomogeneous Poisson process whichimplies in turn that the largest eigenvalue converges in distribution to aFr\'echet distributed random variable. They proved these results under theassumption that $p$ and $n$ are proportional to each other. In this paper weshow that the aforementioned results remain valid if $p$ grows at anypolynomial rate. The proofs are different from those in [2, 31, 32]; we employlarge deviation techniques to achieve them. The proofs reveal that only thediagonal of the sample covariance matrix is relevant for the asymptoticbehavior of the largest eigenvalues and the corresponding eigenvectors whichare close to the canonical basis vectors. We also discuss extensions of theresults to sample autocovariance matrices.

Eigenvalues and eigenvectors of heavy-tailed sample covariance matrices with general growth rates: the iid case