距独立高斯最远的耦合
Couplings Farthest from the Independent Gaussian
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中文总结 AI 辅助
研究联合分布与边际乘积的最大2-Wasserstein距离，证明高斯情形下单调耦合最远，并刻画一般测度的最远分布。
中文摘要 AI 辅助
受Wasserstein依赖性度量的启发，我们研究联合分布与其给定边际乘积之间的最大可能2-Wasserstein距离。对于两个均匀边际，Catalano和Lavenant猜想单调和反单调耦合使与独立耦合的距离最大化。我们证明了对于任意数量$n\geq 2$的一维标准高斯边际，高斯类似物成立。更一般地，对于$\mathbb R$上具有有限二阶矩的每个概率测度$\mu$，我们刻画了$\mathbb{R}^n$上所有边际等于$\mu$且距$n$维标准高斯最远的分布律。
英文摘要
Motivated by Wasserstein measures of dependence, we study the largest possible 2-Wasserstein distance between a joint distribution and the product of its prescribed marginals. For two uniform marginals, Catalano and Lavenant conjectured that the monotone and antimonotone couplings maximize the distance from the independent coupling. We prove the Gaussian analogue for an arbitrary number $n\geq 2$ of one-dimensional standard Gaussian marginals. More generally, for every probability measure $μ$ on $\mathbb R$ with finite second moment, we characterize the laws on $\mathbb{R}^n$ with all marginals equal to $μ$ that are farthest from the $n$-dimensional standard Gaussian.
