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

Probability 2012-10-31 v2 Statistics Theory Statistics Theory

Abstract

We study the joint limit distribution of the kk largest eigenvalues of a p×pp\times p sample covariance matrix XX\TXX^\T based on a large p×np\times n matrix XX. The rows of XX are given by independent copies of a linear process, Xit=jcjZi,tjX_{it}=\sum_j c_j Z_{i,t-j}, with regularly varying noise (Zit)(Z_{it}) with tail index α(0,4)\alpha\in(0,4). It is shown that a point process based on the eigenvalues of XX\TXX^\T converges, as nn\to\infty and pp\to\infty at a suitable rate, in distribution to a Poisson point process with an intensity measure depending on α\alpha and cj2\sum c_j^2. This result is extended to random coefficient models where the coefficients of the linear processes (Xit)(X_{it}) are given by cj(θi)c_j(\theta_i), for some ergodic sequence (θi)(\theta_i), and thus vary in each row of XX. 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/n goes to zero or infinity and α(0,2)\alpha\in(0,2).

Keywords

Cite

@article{arxiv.1108.5464,
  title  = {Limit Theory for the largest eigenvalues of sample covariance matrices with heavy-tails},
  author = {Richard A. Davis and Oliver Pfaffel and Robert Stelzer},
  journal= {arXiv preprint arXiv:1108.5464},
  year   = {2012}
}

Comments

30 pages