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 $mu_{n} = rac {1}{n} sum_{k=1}^n delta_{X_k}$, $n geq 1$, 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 $ ig [ mathbb{E}(W_{p}^p(mu_{n},mu )) ig ]^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 $ rac {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.