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August, 1992 Matching Random Subsets of the Cube with a Tight Control on One Coordinate
WanSoo T. Rhee, Michel Talagrand
Ann. Appl. Probab. 2(3): 695-713 (August, 1992). DOI: 10.1214/aoap/1177005655


Consider a measure $\mu$ on $\lbrack 0, 1\rbrack^2$, and $2n$ points $X_1, \cdots, X_n, Y_1, \cdots, Y_n$ that are independent and distributed according to $\mu$. Consider $2n$ points $U_1, \cdots, U_n, V_1, \cdots, V_n$ that are independent and uniformly distributed on $\lbrack 0, 1 \rbrack$. Then there exists a constant $K$ (independent of $\mu$) such that if $s \leq \sqrt n / K$, with probability close to 1 we can find a one-to-one map $\pi$ from $\{1, \cdots, n\}$ to itself such that $\forall i \leq n, \quad |U_i - V_{\pi(i)}| \leq \frac{K}{s},$ $\frac{1}{n} \sum_{i \leq n} |X_i - Y_{\pi(i)}| \leq K \big(\frac{s}{n}\big)^{1/2}.$


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WanSoo T. Rhee. Michel Talagrand. "Matching Random Subsets of the Cube with a Tight Control on One Coordinate." Ann. Appl. Probab. 2 (3) 695 - 713, August, 1992.


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

zbMATH: 0756.60011
MathSciNet: MR1177905
Digital Object Identifier: 10.1214/aoap/1177005655

Primary: 60D05

Keywords: Gaussian processes , matching problems , Random subsets

Rights: Copyright © 1992 Institute of Mathematical Statistics

Vol.2 • No. 3 • August, 1992
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