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One-dimensional empirical measures, order statistics, and Kantorovich transport distances

About this Title

Sergey Bobkov, School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455 and Michel Ledoux, Institut de Mathématiques de Toulouse, Université de Toulouse, F-31062 Toulouse, France — and — Institut Universitaire de France

Publication: Memoirs of the American Mathematical Society
Publication Year: 2019; Volume 261, Number 1259
ISBNs: 978-1-4704-3650-6 (print); 978-1-4704-5401-2 (online)
DOI: https://doi.org/10.1090/memo/1259
Published electronically: November 5, 2019
Keywords: Empirical measure, Kantorovich distance, rate of convergence, finite rate bound, order statistic, inverse distribution function, beta distribution, log-concave measure.
MSC: Primary 60B10, 60F99, 60G57, 62G30, 60B12; Secondary 62G20

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Table of Contents

Chapters

  • 1. Introduction
  • 2. Generalities on Kantorovich transport distances
  • 3. The Kantorovich distance $W_1(\mu _n, \mu )$
  • 4. Order statistics representations of $W_p(\mu _n, \mu )$
  • 5. Standard rate for ${\mathbb {E}}(W_p^p(\mu _n,\mu ))$
  • 6. Sampling from log-concave distributions
  • 7. Miscellaneous bounds and results

Appendices

  • A. Inverse distribution functions
  • B. Beta distributions

Abstract

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.

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