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 $n^{\mathrm{th}}$ 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 $n^{\mathrm{th}}$ 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 $d\ge 2$ and $n\ge 1$ prove that the image of the mapping $p_n$ on the domain of NRPD (respectively, PRPD) distributions is $[p_n^{\ast },1]$ (resp., $[n^{-1},p_n^{\ast }]$), where $p_n^{\ast }$ is the record-setting probability for any continuous $F$ governing independent coordinates.

16 pages, 1 figure; this revision responds to three anonymous reviews; paper accepted to Probability in the Engineering and Informational Sciences