اندازه‌گیری‌های تجربی یک‌بعدی، آمار مرتب و فاصله‌های حمل و نقل کانتورویچ

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. Cover Title page Chapter 1. Introduction Chapter 2. Generalities on Kantorovich transport distances Chapter 3. The Kantorovich distance W1(μ_{n},μ) Chapter 4. Order statistics representations of W_{p}(μ_{n},μ) Chapter 5. Standard rate for \E(W_{p}^{p}(μ_{n},μ)) 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