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November, 1992 Matching Random Samples in Many Dimensions
Michel Talagrand
Ann. Appl. Probab. 2(4): 846-856 (November, 1992). DOI: 10.1214/aoap/1177005578

Abstract

Consider any norm $N$ on $\mathbb{R}^d, d \geq 3$, and independent uniformly distributed points $X_1, \ldots, X_n, \ldots; Y_1, \ldots, Y_n, \ldots$ in $\lbrack 0, 1\rbrack^d$. Consider the random variable $M_n = \inf \sum_{i \leq n} N(X_i - Y_{\sigma(i)})$, where the infimum is taken over all permutations $\sigma$ of $\{1, \ldots, n\}$. We show that for some universal constant $K$, we have $\lim \sup_{n \rightarrow \infty} M_n n^{-1 + 1/d} \leq r_N \big(1 + K \frac{\log d}{d}\big)mathrm{a,s.},$ where $r_N$ is the radius of the ball for $N$ of volume 1.

Citation

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Michel Talagrand. "Matching Random Samples in Many Dimensions." Ann. Appl. Probab. 2 (4) 846 - 856, November, 1992. https://doi.org/10.1214/aoap/1177005578

Information

Published: November, 1992
First available in Project Euclid: 19 April 2007

zbMATH: 0761.60007
MathSciNet: MR1189420
Digital Object Identifier: 10.1214/aoap/1177005578

Subjects:
Primary: 60C05
Secondary: 05C70

Keywords: empirical measure , Matchings , transportation cost

Rights: Copyright © 1992 Institute of Mathematical Statistics

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Vol.2 • No. 4 • November, 1992
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