The XYZ Conjecture and the FKG Inequality
The XYZ Conjecture and the FKG Inequality
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XYZ 猜想和 FKG 不等式
DOI：
10.1214/aop/1176993791
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发表时间：
1982
影响因子：
2.3
通讯作者：
L. Shepp
中科院分区：
文献类型：
--
作者：
L. Shepp
考虑随机变量\(x_1,\cdots,x_n\)，它们在单位区间上独立且均匀分布。假设我们得到了关于\(x\)的未知排序的部分信息\(\Gamma\)；例如，\(\Gamma = \{x_1 < x_{12}, x_7 < x_5,\cdots\}\)。我们证明了“XYZ猜想”（最初由伊万·里瓦尔、比尔·桑兹提出，并由彼得·温克勒、R. L. 格雷厄姆以及1981年班夫有序集研讨会上的其他参与者扩展），即\(P(x_1 < x_2|\Gamma) \leq P(x_1 < x_2|\Gamma, x_1 < x_3)\)。该证明基于相关性的FKG不等式，并通过实例表明，即使FKG不等式的假设不成立，也有可能重新定义偏序，使得FKG不等式的结论仍然成立。
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<<em>x2|Γ) ≤ P(x1<<em>x2| Γ,x1<<em>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.