论文标题
通过确定性的经验措施，在实际线路上的Wasserstein距离中的近似率
Approximation rate in Wasserstein distance of probability measures on the real line by deterministic empirical measures
论文作者
论文摘要
我们对订单级别的近似值$ρ\ ge 1 $ $ n $确定点的经验措施，我们对带有有限订单$ρ$的概率度量$μ$的索引$ρ\ ge 1 $的近似感兴趣。最小错误收敛到$ 0 $作为$ n \ to+\ infty $，我们试图表征与此收敛相关的订单。在\ cite {Xuberger}中，Xu和Berger表明，当$μ$是零零质量并且错误消失时，该订单不超过$ 1 $，并给出了足够的条件，使该订单与该阈值$ 1 $相等的$ 1 $在与Lebesgue Matues a $ $ $ $ $ $ $ $ $ $ $ $ $的密度方面相同。他们还证明，当$μ$的支撑是有限的，而当支持不是间隔时，订单不小于$ 1/ρ$。我们通过检查命令以$ \左为单位（1/ρ，1 \右）$的命令来补充这些结果，必须对支架进行界限，并说明$μ$的尾巴上的必要和足够条件，以使订单的订单以等于一定的给定值，以便在间隔$ \ weft（0,1/ρ\ right）中，以至于有足够的条件（0,1/ρ\右）。 \ cite {xuberger}。鉴于实际应用，我们强调的是，在证明最小误差收敛顺序的每个结果的证据中，我们根据$μ$的分数功能表现出明确的点选择，这些$μ$表现出相同的收敛顺序。
We are interested in the approximation in Wasserstein distance with index $ρ\ge 1$ of a probability measure $μ$ on the real line with finite moment of order $ρ$ by the empirical measure of $N$ deterministic points. The minimal error converges to $0$ as $N\to+\infty$ and we try to characterize the order associated with this convergence. In \cite{xuberger}, Xu and Berger show that, apart when $μ$ is a Dirac mass and the error vanishes, the order is not larger than $1$ and give a sufficient condition for the order to be equal to this threshold $1$ in terms of the density of the absolutely continuous with respect to the Lebesgue measure part of $μ$. They also prove that the order is not smaller than $1/ρ$ when the support of $μ$ is bounded and not larger when the support is not an interval. We complement these results by checking that for the order to lie in the interval $\left(1/ρ,1\right)$, the support has to be bounded and by stating a necessary and sufficient condition in terms of the tails of $μ$ for the order to be equal to some given value in the interval $\left(0,1/ρ\right)$, thus precising the sufficient condition in terms of moments given in \cite{xuberger}. In view of practical application, we emphasize that in the proof of each result about the order of convergence of the minimal error, we exhibit a choice of points explicit in terms of the quantile function of $μ$ which exhibits the same order of convergence.
