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 |x−y| is smaller than twice their W1-distance (Wasserstein distance with index 1). We showed that replacing |x−y| and W1 respectively with |x−y|ρ and Wρρ does not lead to a finite multiplicative constant. We show here that a finite constant is recovered when replacing Wρρ with the product of Wρ times the centred ρ-th moment of the second marginal to the power ρ−1. Then we study the generalisation of this new martingale Wasserstein inequality to higher dimension.
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