Translator Disclaimer
December 2021 A simple Fourier analytic proof of the AKT optimal matching theorem
Sergey G. Bobkov, Michel Ledoux
Author Affiliations +
Ann. Appl. Probab. 31(6): 2567-2584 (December 2021). DOI: 10.1214/20-AAP1656

Abstract

We present a short and elementary proof of the Ajtai–Komlós–Tusnády (AKT) optimal matching theorem in dimension 2 via Fourier analysis and a smoothing argument. The upper bound applies to more general families of samples, including dependent variables, of interest in the study of rates of convergence for empirical measures. Following the recent pde approach by L. Ambrosio, F. Stra and D. Trevisan, we also adapt a simple proof of the lower bound.

Funding Statement

Research of S.B. was partially supported by NSF Grant DMS-1855575.

Acknowledgments

The authors thank the reviewer for relevant suggestions of improvements in the exposition, and for pointing out a technical issue in one proof.

Citation

Download Citation

Sergey G. Bobkov. Michel Ledoux. "A simple Fourier analytic proof of the AKT optimal matching theorem." Ann. Appl. Probab. 31 (6) 2567 - 2584, December 2021. https://doi.org/10.1214/20-AAP1656

Information

Received: 1 September 2019; Revised: 1 December 2020; Published: December 2021
First available in Project Euclid: 13 December 2021

Digital Object Identifier: 10.1214/20-AAP1656

Subjects:
Primary: 49Q22 , 60D05
Secondary: 49J55 , 58J35 , 60H15 , 62G30

Keywords: Ajtai–Komlós–Tusnády theorem , empirical measure , Fourier analysis , heat kernel smoothing , optimal matching

Rights: Copyright © 2021 Institute of Mathematical Statistics

JOURNAL ARTICLE
18 PAGES

This article is only available to subscribers.
It is not available for individual sale.
+ SAVE TO MY LIBRARY

SHARE
Vol.31 • No. 6 • December 2021
Back to Top