Rates for Bayesian estimation of location-scale mixtures of super-smooth densities

We consider Bayesian nonparametric density estimation with a Dirichlet process kernel mixture as a prior on the class of Lebesgue univariate densities, the emphasis being on the achievability of the error rate $1/\sqrt{n}$, up to a logarithmic factor, depending on the kernel. We derive rates of convergence for the Bayes' estimator of super-smooth densities that are location-scale mixtures of densities whose Fourier transforms have sub-exponential tails. We show that a nearly parametric rate is attainable in $L^1$-norm, under weak assumptions on the tail decay of the true mixing distribution and the overall Dirichlet process base measure.