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

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

2011-08-27

arxiv.org/abs/1108.5464v1

Mathematics

Probability

Scientific paper

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 $\alpha\in(0,2)$. It is shown that the point process based on the eigenvalues of $XX^\T$ converges in distribution to a Poisson point process with intensity measure depending on $\alpha$ 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 $c_j(\theta_i)$, for some ergodic sequence $(\theta_i)$, and thus vary in each row of $X$. As a by-product, we obtain a proof of the corresponding result for matrices with iid entries in cases where $p/n$ goes to zero or infinity.

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