Given a sequence of independent random vectors taking values in ${\mathbb R}^d$ and having common continuous distribution function $F$, say that the $n^{\rm \scriptsize th}$ observation sets a (Pareto) record if it is not dominated (in every coordinate) by any preceding observation. Let $p_n(F) \equiv p_{n, d}(F)$ denote the probability that the $n^{\rm \scriptsize th}$ observation sets a record. There are many interesting questions to address concerning $p_n$ and multivariate records more generally, but this short paper focuses on how $p_n$ varies with $F$, particularly if, under $F$, the coordinates exhibit negative dependence or positive dependence (rather than independence, a more-studied case). We introduce new notions of negative and positive dependence ideally suited for such a study, called negative record-setting probability dependence (NRPD) and positive record-setting probability dependence (PRPD), relate these notions to existing notions of dependence, and for fixed $d \geq 2$ and $n \geq 1$ prove that the image of the mapping $p_n$ on the domain of NRPD (respectively, PRPD) distributions is $[p^*_n, 1]$ (resp., $[n^{-1}, p^*_n]$), where $p^*_n$ is the record-setting probability for any continuous $F$ governing independent coordinates.

翻译：给定一系列取值为${\mathbb R}^d$的独立随机向量，且具有共同的连续分布函数$F$，称第$n$个观测值创造了一个（帕累托）记录，如果它不被任何先前观测值（在每个坐标上）支配。设$p_n(F) \equiv p_{n, d}(F)$表示第$n$个观测值创造记录的概率。关于$p_n$以及更一般的多元记录存在许多有趣的问题需要探讨，但本文聚焦于$p_n$随$F$变化的方式，特别是在$F$下坐标呈现负相关或正相关（而非独立性这一更常研究的情况）时。我们引入了适合此类研究的负相关和正相关新概念，分别称为负记录设定概率依赖（NRPD）和正记录设定概率依赖（PRPD），将这些概念与现有相关概念联系起来，并在固定$d \geq 2$和$n \geq 1$时证明，映射$p_n$在NRPD（分别为PRPD）分布域上的像集为$[p^*_n, 1]$（分别为$[n^{-1}, p^*_n]$），其中$p^*_n$是任何控制独立坐标的连续$F$下创造记录的概率。