On the Stochastic Convergence of Representations Based on Wasserstein Metrics

Abstract

Suppose that $P$ and $P_n, n \in \mathscr{N}$, are probabilities on a real, separable Hilbert space, $V$. It is known that if $P$ satisfies some regularity conditions and $X$ is such that $P_X = P$, then there exist mappings $H_n: V \rightarrow V$, such that $P_{H_n(X)} = P_n$ and the Wasserstein distance between $P_n$ and $P$ coincides with $(\int\|x - H_n(x)\|^2 dP)^{1/2}, n \in \mathscr{N}$. In this paper we prove that the weak convergence of $\{P_n\}$ to $P$ is enough to ensure that $\{H_n(X)\}$ converges to $X$ in measure, and that, if $V = \mathcal{R}^p$, then the convergence is also a.e. This property seems to be characteristic of finite-dimensional spaces, because we include an example, with $V$ infinite-dimensional and $P$ Gaussian, where a.e. convergence does not hold.