Given a sequence of independent random vectors taking values in ${\mathbb R}^d$ and having common continuous distribution $F$, 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 (NRSPD) and positive record-setting probability dependence (PRSPD), 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 NRSPD (respectively, PRSPD) distributions is $[p^*_n, 1]$ (resp., $[n^{-1}, p^*_n]$), where $p^*_n$ is the record-setting probability for any $F$ governing independent coordinates.

翻译：给定一个取值于${\mathbb R}^d$的独立随机向量序列，其共同连续分布为$F$，称第$n$个观测值创造一个（Pareto）记录，若它不被任何前面的观测值（在每个坐标上）所支配。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$p_n$及更一般的多元记录存在许多有趣的问题有待探讨，但本文简短地聚焦于$p_n$如何随$F$变化，特别是当$F$下坐标呈现负相关或正相关时（而非独立性——一种研究较多的情况）。我们引入了非常适合此类研究的新颖负相关和正相关概念，分别称为负记录设定概率依赖（NRSPD）和正记录设定概率依赖（PRSPD），将这些概念与现有相关概念联系起来，并针对固定$d \geq 2$和$n \geq 1$，证明了在NRSPD（分别地，PRSPD）分布定义域上，映射$p_n$的像集为$[p^*_n, 1]$（分别地，$[n^{-1}, p^*_n]$），其中$p^*_n$是任意控制独立坐标的$F$下的记录设定概率。