Limit Distributions and Sensitivity Analysis for Empirical Entropic Optimal Transport on Countable Spaces

TL;DR

This paper establishes the asymptotic distributional limits for empirical entropic optimal transport on countable spaces, showing convergence to Gaussian processes and normality of transport values, with broad applicability and bootstrap consistency.

Contribution

It extends distributional limit results for empirical entropic optimal transport from finite to countable spaces using novel sensitivity analysis and bounds.

Findings

Empirical optimal transport plan converges to a Gaussian process.

Empirical entropic optimal transport value is asymptotically normal.

Bootstrap consistency is established for statistical inference.

Abstract

For probability measures on countable spaces we derive distributional limits for empirical entropic optimal transport quantities. More precisely, we show that the empirical optimal transport plan weakly converges to a centered Gaussian process and that the empirical entropic optimal transport value is asymptotically normal. The results are valid for a large class of cost functions and generalize distributional limits for empirical entropic optimal transport quantities on finite spaces. Our proofs are based on a sensitivity analysis with respect to norms induced by suitable function classes, which arise from novel quantitative bounds for primal and dual optimizers, that are related to the exponential penalty term in the dual formulation. The distributional limits then follow from the functional delta method together with weak convergence of the empirical process in that respective norm, for which we provide sharp conditions on the underlying measures. As a byproduct of our proof technique, consistency of the bootstrap for statistical applications is shown.