Uniform Convergence of Empirical Transport Maps

This note examines the stability property of Brenier's transport (quantile map) $Q$, defined as the gradient of a convex potential that transports a distribution $F$ to a distribution $P$ on $R^d$, i.e. if $U$ is a random vector distributed as $F$, then the random vector $Q(U)$ is distributed as $P$. Specifically, if $\hat F_n$ and $\hat P_n$ are empirical measures converging to $F$ and $P$ in any metric that metrizes weak convergence, then the empirical Brenier transport $\hat Q_n$ transporting $\hat F_n$ to $\hat P_n$ converges uniformly to $Q$ on the compact subsets of the interior of support of $F$. This is useful in economic and statistical analysis, where the transport maps had been used as a form of multivariate, vector-valued quantiles and ranks, e.g (EGH, 2012,CCG, 2014). This note is a part of another project (CGHH, 2013).