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August, 1982 The XYZ Conjecture and the FKG Inequality
L. A. Shepp
Ann. Probab. 10(3): 824-827 (August, 1982). DOI: 10.1214/aop/1176993791


Consider random variables $x_1, \cdots, x_n$, independently and uniformly distributed on the unit interval. Suppose we are given partial information, $\Gamma$, about the unknown ordering of the $x$'s; e.g., $\Gamma = \{x_1 < x_{12}, x_7 < x_5, \cdots\}$. We prove the "XYZ conjecture" (originally due to Ivan Rival, Bill Sands, and extended by Peter Winkler, R. L. Graham, and other participants of the Symposium on Ordered Sets at Banff, 1981) that $P(x_1 < x_2|\Gamma) \leq P(x_1 < x_2|\Gamma, x_1 < x_3).$ The proof is based on the FKG inequality for correlations and shows by example that even when the hypothesis of the FKG inequality fails it may be possible to redefine the partial ordering so that the conclusion of the FKG inequality still holds.


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L. A. Shepp. "The XYZ Conjecture and the FKG Inequality." Ann. Probab. 10 (3) 824 - 827, August, 1982.


Published: August, 1982
First available in Project Euclid: 19 April 2007

zbMATH: 0484.60010
MathSciNet: MR659563
Digital Object Identifier: 10.1214/aop/1176993791

Primary: 05A20
Secondary: 60C05

Keywords: FKG inequality , partially ordered sets , XYZ conjecture

Rights: Copyright © 1982 Institute of Mathematical Statistics

Vol.10 • No. 3 • August, 1982
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