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.
General
Imprint: |
American Mathematical Society
|
Country of origin: |
United States |
Series: |
Memoirs of the American Mathematical Society |
Release date: |
October 2020 |
Authors: |
Sergey Bobkov
• Michel Ledoux
|
Dimensions: |
254 x 178mm (L x W) |
Format: |
Paperback
|
Pages: |
126 |
ISBN-13: |
978-1-4704-3650-6 |
Categories: |
Books
|
LSN: |
1-4704-3650-7 |
Barcode: |
9781470436506 |
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