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One-dimensional empirical measures, order statistics, and Kantorovich transport distances
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One-dimensional empirical measures, order statistics, and Kantorovich transport distances
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One-dimensional empirical measures, order statistics, and Kantorovich transport distances
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One-dimensional empirical measures, order statistics, and Kantorovich transport distances
2019
Overview
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.
