Richard A. Davis

Publications (2)0 Total impact

• Article: Eigenvalues of sample covariance matrices of non-linear processes with infinite variance

  Richard A. Davis, Oliver Pfaffel
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  ABSTRACT: We study the $k$-largest eigenvalues of heavy-tailed sample covariance matrices of the form $\bX\bX^\T$ in an asymptotic framework, where the dimension of the data and the sample size tend to infinity. To this end, we assume that the rows of $\bX$ are given by independent copies of some stationary process with regularly varying marginals with index $\alpha\in(0,2)$ satisfying large deviation and mixing conditions. We apply these general results to stochastic volatility and GARCH processes.
  11/2012;
• Source

  Available from: ArXiv

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

  Richard A. Davis, Oliver Pfaffel, Robert Stelzer
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  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 $\alpha\in(0,2)$.
  08/2011;