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.
Funding Statement
S. Hundrieser acknowledges funding by the Deutsche Forschungsgemeinschaft under Germany’s Excellence Strategy—EXC 2067/1-390729940 and under RTG 2088.
M. Klatt acknowledges funding by the Deutsche Forschungsgemeinschaft DFG RTG 2088.
A. Munk acknowledges funding by the Deutsche Forschungsgemeinschaft under Germany’s Excellence Strategy—EXC 2067/1-390729940 and under CRC 1456, A4,C6.
Acknowledgments
The authors would like to thank two anonymous referees and the Editor for their constructive comments that improved the quality of this paper.
Citation
Shayan Hundrieser. Marcel Klatt. Axel Munk. "Limit distributions and sensitivity analysis for empirical entropic optimal transport on countable spaces." Ann. Appl. Probab. 34 (1B) 1403 - 1468, February 2024. https://doi.org/10.1214/23-AAP1995
Information