Asymptotics of Kantorovich Distance for Empirical Measures of the Laguerre Model

TL;DR

This paper analyzes how quickly the Kantorovich distance between empirical measures and their true distribution converges in the Laguerre model, extending known results beyond Gaussian cases.

Contribution

It provides sharp convergence rates for empirical measures in the Laguerre model, a non-compact, asymmetric setting, generalizing previous Gaussian-based results.

Findings

Convergence rates depend on the Laguerre parameter and dimension.

Results are sharp in a specific parameter regime.

Extends Wasserstein distance analysis beyond Gaussian models.

Abstract

We estimate the rate of convergence for the Kantorovich (or Wasserstein) distance between empirical measures of i.i.d. random variables associated with the Laguerre model of order $\alpha$ on $(0,\infty)^N$ and their common law, which is not compactly supported and has no rotational symmetry. Compared with the Gaussian case, our result is sharp provided the parameter $\alpha$ and the dimension $N$ are chosen in a specified regime.