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Full Description
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.
Contents
Introduction
Generalities on Kantorovich transport distances
The Kantorovich distance $W_1(\mu _n, \mu )$
Order statistics representations of $W_p(\mu _n, \mu )$
Standard rate for $\mathbb{E} (W_p^p(\mu _n,\mu ))$
Sampling from log-concave distributions
Miscellaneous bounds and results
Appendix A. Inverse distribution functions
Appendix B. Beta distributions
Bibliography.
