Wasserstein convergence rates of increasingly concentrating probability measures

Abstract

For $\ell :{\mathbb{R}}^d\to [0,\infty )$ we consider the sequence of probability measures ${\left({\mu }_n\right)}_{n\in \mathbb{N}}$, where ${\mu }_n$ is determined by a density that is proportional to $\exp (-n\ell )$. We allow for infinitely many global minimal points of $\ell$, as long as they form a finite union of compact manifolds. In this scenario, we show estimates for the $p$-Wasserstein convergence of ${\left({\mu }_n\right)}_{n\in \mathbb{N}}$ to its limit measure. Imposing regularity conditions we obtain a speed of convergence of $n^{-1/(2p)}$ and adding a further technical assumption, we can improve this to a $p$-independent rate of $1/2$ for all orders $p\in \mathbb{N}$ of the Wasserstein distance.

29 pages, 3 Figures, accepted for publication in Ann. Appl. Probab

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