One-Dimensional Empirical Measures, Order Statistics, and Kantorovich Transport Distances

This work is devoted to the study of rates of convergence of the empirical measures $\mu_{n} = \frac {1}{n} \sum_{k=1}^n \delta_{X_k}$, $n \geq 1$, over a sample $(X_{k})_{k \geq 1}$ of independent identically distributed real-valued random variables towards the common distribution $\mu$ in Kantorovich transport distances $W_p$. The focus is on finite range bounds on the expected Kantorovich distances $\mathbb{E}(W_{p}(\mu_{n},\mu ))$ or $\big [ \mathbb{E}(W_{p}^p(\mu_{n},\mu )) \big ]^1/p$ in terms of moments and analytic conditions on the measure $\mu $ and its distribution function. The study describes a variety of rates, from the standard one $\frac {1}{\sqrt n}$ to slower rates, and both lower and upper-bounds on $\mathbb{E}(W_{p}(\mu_{n},\mu ))$ for fixed $n$ in various instances. Order statistics, reduction to uniform samples and analysis of beta distributions, inverse distribution functions, log-concavity are main tools in the investigation. Two detailed appendices collect classical and some new facts on inverse distribution functions and beta distributions and their densities necessary to the investigation.