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

On concentration of the empirical measure for radial transport costs