Negative dependence of sequences of random variables is often an interesting characteristic of their distribution, as well as a useful tool for studying various asymptotic results, including central limit theorems, Poisson approximations, the rate of increase of the maximum, and more. In the study of probability models of tournaments, negative dependence of participants' outcomes arises naturally with application to various asymptotic results. In particular, the property of negative orthant dependence was proved in several articles for different tournament models, with a special proof for each model. In this note we unify these results by proving a stronger property, negative association, a generalization leading to a very simple proof. We also present a natural example of a knockout tournament where the scores are negatively orthant dependent but not negatively associated. The proof requires a new result on a preservation property of negative orthant dependence that is of independent interest.
翻译:随机变量序列的负依赖性往往是其分布的有趣特征,也是研究各种无症状结果的有用工具,包括中央限值理论、Poisson近似值、最大增长率等等。在对比赛概率模型的研究中,对参与者结果的负依赖性自然会随着对各种无症状结果的应用而产生。特别是,对不同比赛模型的若干文章都证明了负依赖性或超依赖性的性质,对每种模型都有特别的证明。在本说明中,我们通过证明一种更强大的属性、负关联性、普遍化导致非常简单的证明来统一这些结果。我们还举了一个击倒比赛的自然例子,因为得分是负依赖性或超依赖性,但并不与负相关。证据要求对具有独立利益的负依赖性或超依赖性保护性的财产产生新的结果。