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

Abstract

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 $\alpha \in (0,4)$. 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 $\alpha$ 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 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 $\alpha \in (0,2)$.

30 pages

References in corpus (1)

• Eigenvalue distribution of large sample covariance matrices of linear processes

Cited by in corpus (2)

• On the spectral norm of large heavy-tailed random matrices with strongly dependent rows and columns
• Eigenvalues of sample covariance matrices of non-linear processes with infinite variance