  This paper discusses fluctuations of linear spectral statistics of
high-dimensional sample covariance matrices when the underlying population
follows an elliptical distribution. Such population often possesses high order
correlations among their coordinates, which have great impact on the asymptotic
behaviors of linear spectral statistics. Taking such kind of dependency into
consideration, we establish a new central limit theorem for the linear spectral
statistics in this paper for a class of elliptical populations. This general
theoretical result has wide applications and, as an example, it is then applied
to test the sphericity of elliptical populations.
