Peer-reviewedCC BYProbability Theory and Related Fields
Sharp inequalities between Zolotarev and Wasserstein distances in 𝒫2(ℝd)
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 ≥ 1 d ≥ 1 with an optimal constant. Similarly, we propose an optimal upper bound for the ratio of Zolotarev distance Z2(μ ,ν ) Z 2 ( μ , ν ) to Wasserstein distance W2(μ ,ν ) W 2 ( μ , ν ) when μ ,ν ∈ 𝒫2(ℝd) μ , ν ∈ P 2 ( R d ) are centred probabilities with prescribed variances.
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BibTeX
@article{bolbotowski2026sharp,
title = {{Sharp inequalities between Zolotarev and Wasserstein distances in $$\mathcal P_2(\mathbb R^d)$$}},
author = {Karol Bołbotowski and Guy Bouchitté},
journal = {Probability Theory and Related Fields},
year = {2026},
month = jul,
doi = {10.1007/s00440-026-01527-4},
url = {https://hal.science/hal-05846473v1},
note = {CC BY},
}
RIS
TY - JOUR
TI - Sharp inequalities between Zolotarev and Wasserstein distances in 𝒫_2(ℝ^d)
AU - Karol Bołbotowski
AU - Guy Bouchitté
PY - 2026
DA - 2026/07/30
JO - Probability Theory and Related Fields
DO - 10.1007/s00440-026-01527-4
UR - https://hal.science/hal-05846473v1
AB - 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 ≥ 1 d ≥ 1 with an optimal constant. Similarly, we propose an optimal upper bound for the ratio of Zolotarev distance Z_2(μ ,ν ) Z 2 ( μ , ν ) to Wasserstein distance W_2(μ ,ν ) W 2 ( μ , ν ) when μ ,ν ∈ 𝒫_2(ℝ^d) μ , ν ∈ P 2 ( R d ) are centred probabilities with prescribed variances.
SN - 0178-8051
SN - 1432-2064
ER -