Multivariate Log-Concave Distributions as a Nearly Parametric Model

TL;DR

This paper demonstrates that multivariate log-concave distributions form a nearly parametric model, with strong convergence properties similar to parametric families like Gaussian distributions, despite being nonparametric.

Contribution

It establishes that weak convergence in the family of log-concave distributions implies stronger forms of convergence, highlighting their nearly parametric nature.

Findings

Weak convergence implies total variation convergence.

Arbitrary moments converge under weak convergence.

Laplace transforms converge pointwise.

Abstract

In this paper we show that the family P_d of probability distributions on R^d with log-concave 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. Hence the nonparametric model P_d has similar properties as parametric models such as, for instance, the family of all d-variate Gaussian distributions.