Quantified Cram\'er-Wold Continuity Theorem for the Kantorovich Transport Distance

TL;DR

This paper provides a quantitative version of the Cramér-Wold theorem, establishing an upper bound for the Kantorovich transport distance between high-dimensional probability measures based on their one-dimensional projections, thus advancing understanding of measure convergence.

Contribution

It introduces a quantified Cramér-Wold continuity theorem that bounds the Kantorovich transport distance using one-dimensional projections, offering a new tool for analyzing measure convergence.

Findings

Provides an explicit upper bound for the Kantorovich distance

Quantifies the Cramér-Wold theorem for weak convergence

Enhances methods for comparing high-dimensional probability measures

Abstract

An upper bound for the Kantorovich transport distance between probability measures on multidimensional Euclidean spaces is given in terms of transport distances between one dimensional projections. This quantifies the Cram\'er-Wold continuity theorem for the weak convergence of probability measures.