  This paper investigates limiting properties of eigenvalues of multivariate
sample spatial-sign covariance matrices when both the number of variables and
the sample size grow to infinity. The underlying p-variate populations are
general enough to include the popular independent components model and the
family of elliptical distributions. A first result of the paper establishes
that the distribution of the eigenvalues converges to a deterministic limit
that belongs to the family of generalized Marcenko-Pastur distributions.
Furthermore, a new central limit theorem is established for a class of linear
spectral statistics. We develop two applications of these results to robust
statistics for a high-dimensional shape matrix. First, two statistics are
proposed for testing the sphericity. Next, a spectrum-corrected estimator using
the sample spatial-sign covariance matrix is proposed. Simulation experiments
show that in high dimension, the sample spatial-sign covariance matrix provides
a valid and robust tool for mitigating influence of outliers.
