One-dimensional empirical measures, order statistics, and Kantorovich transport distances
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 tran...
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| Format: | UnknownFormat |
| Sprache: | eng |
| Veröffentlicht: |
Providence, RI
American Mathematical Society
2019
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| Schriftenreihe: | Memoirs of the American Mathematical Society
volume 261, number 1259 (September 2019) |
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| Online Zugang: | Inhaltsverzeichnis |
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Tag hinzufügen | Zusammenfassung: | 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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| Beschreibung: | "September 2019, volume 261, number 1259 (third of 7 numbers)" Literaturverzeichnis: Seite 121-126 |
| Beschreibung: | v, 126 Seiten Illustrationen |
| ISBN: | 9781470436506 978-1-4704-3650-6 |