Statistics Papers

Document Type

Journal Article

Date of this Version

1982

Publication Source

The Annals of Probability

Volume

10

Issue

3

Start Page

824

Last Page

827

DOI

10.1214/aop/1176993791

Abstract

Consider random variables x1,⋯,xn, independently and uniformly distributed on the unit interval. Suppose we are given partial information, Γ, about the unknown ordering of the x's; e.g., Γ = {x1 < x12, x7 < x5,⋯}. 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(x1<x2|Γ) ≤ P(x1<x2| Γ,x1<x3).

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.

Keywords

FKG inequality, XYZ conjecture, partially ordered sets

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Date Posted: 27 November 2017

This document has been peer reviewed.