By studying the family of $p$-dimensional scale mixtures, this paper shows for the first time a non trivial example where the eigenvalue distribution of the corresponding sample covariance matrix {em does not converge} to the celebrated Marv{c}enko-Pastur law. A different and new limit is found and characterized. The reasons of failure of the Marv{c}enko-Pastur limit in this situation are found to be a strong dependence between the $p$-coordinates of the mixture. Next, we address the problem of testing whether the mixture has a spherical covariance matrix. To analize the traditional Johns type test we establish a novel and general CLT for linear statistics of eigenvalues of the sample covariance matrix. It is shown that the Johns test and its recent high-dimensional extensions both fail for high-dimensional mixtures, precisely due to the different spectral limit above. As a remedy, a new test procedure is constructed afterwards for the sphericity hypothesis. This test is then applied to identify the covariance structure in model-based clustering. It is shown that the test has much higher power than the widely used ICL and BIC criteria in detecting non spherical component covariance matrices of a high-dimensional mixture.
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