On the probability of a Pareto record

Abstract

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 pn and multivariate records more generally, but this short paper focuses on how pn 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 pn 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.