Quantified Cramér-Wold Continuity Theorem for the Kantorovich Transport Distance

TL;DR

The paper provides a quantitative version of the Cramér-Wold continuity theorem for the Kantorovich transport distance $W$, establishing a dimension- and moment-dependent bound $W(X,Y) \le 18\, b^{1-\alpha}\sup_{|\theta|=1}W(X_\theta,Y_\theta)^\alpha$ with $\alpha=\frac{2}{d p^*+2}$ under $p>1$ moment bounds. It shows that high-dimensional weak convergence can be controlled via one-dimensional projections and develops a robust analytic framework combining truncation, smoothing, Fourier analysis, and Kantorovich duality. The work also analyzes empirical projections, proving $\mathbb{E}\sup_{|\theta|=1} W(\mu_{n,\theta},\mu_\theta) \le \frac{c_d}{\sqrt{n}}$ and deriving asymptotic optimality of the exponent $\alpha$ (no larger than $2/d$) through rates for $\mathbb{E}W(\mu_n,\mu)$. The results hinge on a delicate blend of truncation, Plancherel-type arguments, and chaining, providing a principled pathway to reduce multidimensional transport problems to one-dimensional analysis.

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 Cramér-Wold continuity theorem for the weak convergence of probability measures.