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Softcover ISBN:  9781470436506 
Product Code:  MEMO/261/1259 
List Price:  $81.00 
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eBook ISBN:  9781470454012 
Product Code:  MEMO/261/1259.E 
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AMS Member Price:  $48.60 
Softcover ISBN:  9781470436506 
eBook ISBN:  9781470454012 
Product Code:  MEMO/261/1259.B 
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MAA Member Price:  $145.80 $109.35 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 261; 2019; 126 ppMSC: Primary 60; 62
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 realvalued 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 upperbounds 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, logconcavity 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.

Table of Contents

Chapters

1. Introduction

2. Generalities on Kantorovich transport distances

3. The Kantorovich distance $W_1(\mu _n, \mu )$

4. Order statistics representations of $W_p(\mu _n, \mu )$

5. Standard rate for ${\mathbb {E}}(W_p^p(\mu _n,\mu ))$

6. Sampling from logconcave distributions

7. Miscellaneous bounds and results

Appendices

A. Inverse distribution functions

B. Beta distributions


Additional Material

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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 realvalued 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 upperbounds 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, logconcavity 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.

Chapters

1. Introduction

2. Generalities on Kantorovich transport distances

3. The Kantorovich distance $W_1(\mu _n, \mu )$

4. Order statistics representations of $W_p(\mu _n, \mu )$

5. Standard rate for ${\mathbb {E}}(W_p^p(\mu _n,\mu ))$

6. Sampling from logconcave distributions

7. Miscellaneous bounds and results

Appendices

A. Inverse distribution functions

B. Beta distributions