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One-Dimensional Empirical Measures, Order Statistics, and Kantorovich Transport Distances (Memoirs of the American Mathematical Society, September 2019, Number 1259) free ebook download

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Author(s):	Sergey Bobkov

Date:	2024	Format:	pdf	Language:	English	ISBN/ASIN:	1470436507
Pages:	126	OCR:		Quality:		ISBN13:
Uploader:	maveriks	Upload Date:	2/12/2024 2:22:28 AM

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This work is devoted to the study of rates of convergence of the empirical measures $mu_{n} = rac {1}{n} sum_{k=1}^n delta_{X_k}$, $n geq 1$, 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 $ ig [ mathbb{E}(W_{p}^p(mu_{n},mu )) ig ]^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 $ rac {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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Tags: American Distances Empirical Kantorovich Measures Memoirs One-Dimensional Order Statistics Transport

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One-Dimensional Empirical Measures, Order Statistics, and Kantorovich Transport Distances (Memoirs of the American Mathematical Society, September 2019, Number 1259) free ebook download

Category: Mathematics

Description: Click to see full description

Category:
Mathematics