On the asymptotics of Ajtai-Koml\'os-Tusn\'ady statistics

TL;DR

This paper investigates the asymptotic behavior of Ajtai-Komlós-Tusnády statistics related to Wasserstein distances, combining computer simulations and theoretical insights, and suggests the limit distribution is Gaussian.

Contribution

It provides the first detailed analysis of the asymptotics of these statistics using simulations and theoretical considerations.

Findings

Simulations indicate a Gaussian limit distribution.

Theoretical analysis supports the Gaussian hypothesis.

Enhanced understanding of Wasserstein distance asymptotics.

Abstract

In our days there is a widespread analysis of Wasserstein distances between theoretical and empirical measures. One of the first investigation of the topic is given in the paper written by Ajtai, Koml\'os and Tusn\'ady in $1984.$ Interestingly, all the neighboring questions posed by that paper were settled already without the original one. In this paper we are going to delineate the limit behavior of the original statistics with the help of computer simulations. At the same time we kept an eye on theoretical grasping of the problem. Based on our computer simulations our opinion is that the limit distribution is Gaussian.