  Density estimation represents one of the most successful applications of
Bayesian nonparametrics. In particular, Dirichlet process mixtures of normals
are the gold standard for density estimation and their asymptotic properties
have been studied extensively, especially in the univariate case. However a gap
between practitioners and the current theoretical literature is present. So
far, posterior asymptotic results in the multivariate case are available only
for location mixtures of Gaussian kernels with independent prior on the common
covariance matrix, while in practice as well as from a conceptual point of view
a location-scale mixture is often preferable. In this paper we address
posterior consistency for such general mixture models by adapting a convergence
rate result which combines the usual low-entropy, high-mass sieve approach with
a suitable summability condition. Specifically, we establish consistency for
Dirichlet process mixtures of Gaussian kernels with various prior
specifications on the covariance matrix. Posterior convergence rates are also
discussed.
