Martingale Wasserstein inequality for probability measures in the convex order

2020/11/23 by Benjamin Jourdain, Jourdain, Benjamin, William Margheriti +1
Mathematics · #Advanced Banach Space Theory #FOS: Mathematics #Geometric Analysis and Curvature Flows #Point processes and geometric inequalities #Probability (math.PR)

paper · doi:10.48550/arxiv.2011.11599

openalex publication_date 2020/11/23 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28

Abstract

It was shown by the authors that two one-dimensional probability measures in the convex order admit a martingale coupling with respect to which the integral of \vert x-y\vert is smaller than twice their \mathcal W₁-distance (Wasserstein distance with index 1). We showed that replacing \vert x-y\vert and \mathcal W₁ respectively with \vert x-y\vert^ρ and \mathcal W_ρ^ρ does not lead to a finite multiplicative constant. We show here that a finite constant is recovered when replacing \mathcal W_ρ^ρ with the product of \mathcal W_ρ times the centred ρ-th moment of the second marginal to the power ρ-1. Then we study the generalisation of this new stability inequality to higher dimension.

Related

• Sampling of probability measures in the convex order by Wasserstein projection
• Quantization and martingale couplings
• Lipschitz continuity of the Wasserstein projections in the convex order on the line
• A new family of one dimensional martingale couplings
• Maximal Martingale Wasserstein Inequality