Let $\mu$ be a probability measure on $\mathbb{R}^d$ and $\mu_N$ its empirical measure with sample size $N$. We prove a concentration inequality for the optimal transport cost between $\mu$ and $\mu_N$ for radial cost functions with polynomial local growth, that can have superpolynomial global growth. This result generalizes and improves upon estimates of Fournier and Guillin. The proof combines ideas from empirical process theory with known concentration rates for compactly supported $\mu$. By partitioning $\mathbb{R}^d$ into annuli, we infer a global estimate from local estimates on the annuli and conclude that the global estimate can be expressed as a sum of the local estimate and a mean-deviation probability for which efficient bounds are known.

翻译：设$\mu$为$\mathbb{R}^d$上的概率测度，$\mu_N$为其样本量为$N$的经验测度。本文针对具有多项式局部增长（允许超多项式全局增长）的径向成本函数，证明了$\mu$与$\mu_N$之间最优运输成本的集中不等式。该结果推广并改进了Fournier与Guillin的估计。证明过程结合了经验过程理论与紧支撑测度已知集中速率的思想。通过将$\mathbb{R}^d$划分为同心环带，我们从环带上的局部估计推导出全局估计，并得出结论：全局估计可表示为局部估计与均值偏差概率之和，其中均值偏差概率的有效界已知。