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

27 August 2011
Richard A. Davis
Oliver Pfaffel
R. Stelzer
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Abstract

We study the joint limit distribution of the kkk largest eigenvalues of a p×pp\times pp×p sample covariance matrix XX\TXX^\TXX\T based on a large p×np\times np×n matrix XXX. The rows of XXX are given by independent copies of a linear process, Xit=∑jcjZi,t−jX_{it}=\sum_j c_j Z_{i,t-j}Xit​=∑j​cj​Zi,t−j​, with regularly varying noise (Zit)(Z_{it})(Zit​) with tail index α∈(0,4)\alpha\in(0,4)α∈(0,4). It is shown that a point process based on the eigenvalues of XX\TXX^\TXX\T converges, as n→∞n\to\inftyn→∞ and p→∞p\to\inftyp→∞ at a suitable rate, in distribution to a Poisson point process with an intensity measure depending on α\alphaα and ∑cj2\sum c_j^2∑cj2​. This result is extended to random coefficient models where the coefficients of the linear processes (Xit)(X_{it})(Xit​) are given by cj(θi)c_j(\theta_i)cj​(θi​), for some ergodic sequence (θi)(\theta_i)(θi​), and thus vary in each row of XXX. As a by-product of our techniques we obtain a proof of the corresponding result for matrices with iid entries in cases where p/np/np/n goes to zero or infinity and α∈(0,2)\alpha\in(0,2)α∈(0,2).

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