Limit Theory for the largest eigenvalues of sample covariance matrices with heavy-tails

We study the joint limit distribution of the $k$ largest eigenvalues of a $p\times p$ sample covariance matrix $XX^\T$ based on a large $p\times n$ matrix $X$. The rows of $X$ are given by independent copies of a linear process, $X_{it}=\sum_j c_j Z_{i,t-j}$, with regularly varying noise $(Z_{it})$ with tail index $α\in(0,4)$. It is shown that a point process based on the eigenvalues of $XX^\T$ converges, as $n\to\infty$ and $p\to\infty$ at a suitable rate, in distribution to a Poisson point process with an intensity measure depending on $α$ and $\sum c_j^2$. This result is extended to random coefficient models where the coefficients of the linear processes $(X_{it})$ are given by $c_j(θ_i)$, for some ergodic sequence $(θ_i)$, and thus vary in each row of $X$. As a by-product of our techniques we obtain a proof of the corresponding result for matrices with iid entries in cases where $p/n$ goes to zero or infinity and $α\in(0,2)$.

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