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This paper discusses fluctuations of linear spectral statistics ofhigh-dimensional sample covariance matrices when the underlying populationfollows an elliptical distribution. Such population often possesses high ordercorrelations among their coordinates, which have great impact on the asymptoticbehaviors of linear spectral statistics. Taking such kind of dependency intoconsideration, we establish a new central limit theorem for the linear spectralstatistics in this paper for a class of elliptical populations. This generaltheoretical result has wide applications and, as an example, it is then appliedto test the sphericity of elliptical populations.

High-dimensional covariance matrices in elliptical distributions with application to spherical test