  This work considers the asymptotic behavior of the distance between two
sample covariance matrices (SCM). A general result is provided for a class of
functionals that can be expressed as sums of traces of functions that are
separately applied to each covariance matrix. In particular, this class
includes very conventional metrics, such as the Euclidean distance or
Jeffrery's divergence, as well as a number of other more sophisticated
distances recently derived from Riemannian geometry considerations, such as the
log-Euclidean metric. In particular, we analyze the asymptotic behavior of this
class of functionals by establishing a central limit theorem that allows us to
describe their asymptotic statistical law. In order to account for the fact
that the sample sizes of two SCMs are of the same order of magnitude as their
observation dimension, results are provided by assuming that these parameters
grow to infinity while their quotients converge to fixed quantities. Numerical
results illustrate how this type of result can be used in order to predict the
performance of these metrics in practical machine learning algorithms, such as
clustering of SCMs.
