  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 Mar\v{c}enko-Pastur law. A different and new limit is found and
characterized. The reasons of failure of the Mar\v{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 John's type test we
establish a novel and general CLT for linear statistics of eigenvalues of the
sample covariance matrix. It is shown that the John's 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.
