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

Cover -- Title page -- Chapter 1. Introduction -- Chapter 2. Generalities on Kantorovich transport distances -- Chapter 3. The Kantorovich distance ₁( _{ }, ) -- Chapter 4. Order statistics representations of _{ }( _{ }, ) -- Chapter 5. Standard rate for \E( _{ }^{ }( _{ }, )) -- Chapter 6. Sampling from log-concave distributions -- Chapter 7. Miscellaneous bounds and results -- Appendices -- Appendix A. Inverse distribution functions -- Appendix B. Beta distributions -- Bibliography -- Back Cover.

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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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.