The Annals of Probability

The Transportation Cost from the Uniform Measure to the Empirical Measure in Dimension $\geq 3$

M. Talagrand

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Abstract

Consider two independent sequences $(X_i)_{i\leq n}$ and $(X'_i)_{i\leq n}$ that are independent and uniformly distributed over $\lbrack 0, 1\rbrack^d, d \geq 3$. Under mild regularity conditions, we describe the convex functions $\varphi$ such that, with large probability, there exists a one-to-one map $\pi$ from $\{1,\ldots,n\}$ to $\{1,\ldots,n\}$ for which $\sum_{i\leq n}\frac{1}{n}\varphi\big(\frac{X_i - X'_{\pi(i)}}{n^{-1/d}K_\varphi}\big) \leq 1,$ where $K_\varphi$ depends on $\varphi$ only.

Article information

Source
Ann. Probab., Volume 22, Number 2 (1994), 919-959.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176988735

Digital Object Identifier
doi:10.1214/aop/1176988735

Mathematical Reviews number (MathSciNet)
MR1288137

Zentralblatt MATH identifier
0809.60015

JSTOR
links.jstor.org

Subjects
Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]

Keywords
Optimal matchings transportation cost empirical measure

Citation

Talagrand, M. The Transportation Cost from the Uniform Measure to the Empirical Measure in Dimension $\geq 3$. Ann. Probab. 22 (1994), no. 2, 919--959. doi:10.1214/aop/1176988735. https://projecteuclid.org/euclid.aop/1176988735


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