Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Sharp inequalities between Zolotarev and Wasserstein distances in $\mathrm{P}_2(\mathbb{R}^d)$

Karol Bołbotowski, Guy Bouchitté

TL;DR

The article establishes sharp, dimension-free bounds linking the Zolotarev distance $Z_2$ and the Wasserstein distance $W_2$ on $\mathcal P_2(\mathbb{R}^d)$. It leverages a newly developed second-order Kantorovich-Rubinstein duality for the Hessian, recasting $Z_2$ as a three-marginal optimal transport problem with martingale and convex-order constraints, and derives a novel auxiliary identity to facilitate bounds. The authors prove a best-possible lower bound $Z_2(\mu,\nu) \ge \tfrac{1}{4} W_2^2(\mu,\nu)$ for equal barycenters, with equality only when $\mu=\nu$, and show no finite reverse bound exists in general. They also prove tight upper bounds $Z_2(\mu,\nu) \le \tfrac{1}{2}(\sigma_\mu + \sigma_\nu) W_2(\mu,\nu)$ (optimal) and a variant involving variances, yielding an equivalence of $W_2$ and $Z_2$ topologies on centered $\mathcal P_2$. Overall, the work sharpens understanding of the relationship between these two fundamental probability metrics and provides tools potentially useful for multivariate CLT rates and related probabilistic approximations.

Abstract

Based on a new Kantorovich-Rubinstein duality principle for the Hessian that was recently established by the two authors, we extend the Rio inequality to any dimension $d \ge 1$ with an optimal constant. Similarly, we propose an optimal upper bound for the ratio of Zolotarev distance $Z_2(μ,ν)$ to Wasserstein distance $W_2(μ,ν)$ when $μ,ν\in \mathrm{P}_2(\mathbb{R}^d)$ are centred probabilities with prescribed variances.

Sharp inequalities between Zolotarev and Wasserstein distances in $\mathrm{P}_2(\mathbb{R}^d)$

TL;DR

The article establishes sharp, dimension-free bounds linking the Zolotarev distance and the Wasserstein distance on . It leverages a newly developed second-order Kantorovich-Rubinstein duality for the Hessian, recasting as a three-marginal optimal transport problem with martingale and convex-order constraints, and derives a novel auxiliary identity to facilitate bounds. The authors prove a best-possible lower bound for equal barycenters, with equality only when , and show no finite reverse bound exists in general. They also prove tight upper bounds (optimal) and a variant involving variances, yielding an equivalence of and topologies on centered . Overall, the work sharpens understanding of the relationship between these two fundamental probability metrics and provides tools potentially useful for multivariate CLT rates and related probabilistic approximations.

Abstract

Based on a new Kantorovich-Rubinstein duality principle for the Hessian that was recently established by the two authors, we extend the Rio inequality to any dimension with an optimal constant. Similarly, we propose an optimal upper bound for the ratio of Zolotarev distance to Wasserstein distance when are centred probabilities with prescribed variances.

Paper Structure

This paper contains 7 sections, 6 theorems, 47 equations.

Table of Contents

  1. introduction
  2. Preliminary results
  3. The feasible set $\Sigma(\mu,\nu)$
  4. The duality result
  5. An auxiliary identity
  6. The lower bound inequality
  7. Upper bound inequalities

Key Result

Lemma 2.2

Let $\pi\in \mathcal{P}_2(({\mathbb{R}^d})^3)$ be a $3$-plan with marginals $(\mu,\nu,\rho)$. Define the marginals $\pi_{1,3} : =\Pi_{1,3}^\#(\pi)$ and $\pi_{2,3} :=\Pi_{2,3}^\#(\pi)$, which are the push forwards of $\pi(dxdydz)$ through the projection maps $(x,y,z)\mapsto (x,z)$ and $(x,y,z)\mapsto Accordingly, an element $\rho\in\mathcal{P}_2({\mathbb{R}^d})$ is the third marginal of a 3-plan $\

Theorems & Definitions (17)

  • Remark 2.1
  • Lemma 2.2
  • Theorem 2.3
  • proof
  • Remark 2.4
  • Lemma 2.5
  • proof
  • Theorem 3.1
  • proof : Proof of the lower bound
  • Example 3.2
  • ...and 7 more