On concentration of the empirical measure for radial transport costs

Understand

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 $\mu$.