Kolmogorov-Smirnov distance and discrepancies versus Wasserstein distances
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866918484403814400 |
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| author | Pagès, Gilles Panloup, Fabien |
| author_facet | Pagès, Gilles Panloup, Fabien |
| contents | We establish inequalities that compare the p-Wasserstein distance to distances which are built as suprema of box measures. More precisely, when the measures are supported on $[0,1]^d$, we obtain sharp upper-bounds of the $p$-Wasserstein distance by (powers of) the (uniform) discrepancy. As an application, we retrieve the Pro\''inov Theorem. When the two distributions are supported {by the whole} $R^d$, {their} $p$-Wasserstein distance is upper bounded by the product of a (power of) their Kolmogorov-Smirnov (KS) distance with the sum of their $p$-moments. Reverse inequalities are established when one of the two distributions has a density, depending on its ${\cal L}^s$-integrability with respect to the Lebesgue measure for some $s>1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_03528 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Kolmogorov-Smirnov distance and discrepancies versus Wasserstein distances Pagès, Gilles Panloup, Fabien Probability We establish inequalities that compare the p-Wasserstein distance to distances which are built as suprema of box measures. More precisely, when the measures are supported on $[0,1]^d$, we obtain sharp upper-bounds of the $p$-Wasserstein distance by (powers of) the (uniform) discrepancy. As an application, we retrieve the Pro\''inov Theorem. When the two distributions are supported {by the whole} $R^d$, {their} $p$-Wasserstein distance is upper bounded by the product of a (power of) their Kolmogorov-Smirnov (KS) distance with the sum of their $p$-moments. Reverse inequalities are established when one of the two distributions has a density, depending on its ${\cal L}^s$-integrability with respect to the Lebesgue measure for some $s>1$. |
| title | Kolmogorov-Smirnov distance and discrepancies versus Wasserstein distances |
| topic | Probability |
| url | https://arxiv.org/abs/2605.03528 |