Multivariate Log-Concave Distributions as a Nearly Parametric Model∗

In this paper we show that the family P d of probability distributions on R d with logconcave densities satisfies a strong continuity condition. In particular, it turns out that weak convergence within this family entails (i) convergence in total variation distance, (ii) convergence of arbitrary moments, and (iii) pointwise convergence of Laplace transforms. In this and several other respects the nonparametric model P d behaves like a parametric model such as, for instance, the family of all d-variate Gaussian distributions. As a consequence of the continuity result, we prove the existence of nontrivial confidence sets for the moments of an unknown distribution in P d . Our results are based on various new inequalities for log-concave distributions which are of independent interest.