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 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，$\mu_N$ 为其样本容量为 $N$ 的经验度量。我们针对具有多项式局部增长且可能具有超多项式全局增长的成本函数，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 和 Guillin 的估计。证明结合了经验过程理论与紧支撑 $\mu$ 的已知集中速率。通过将 $\mathbb{R}^d$ 划分成环状区域，我们从环状区域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已知有效边界的均值偏差概率之和。