On the probability of a Pareto record
James Allen Fill, Ao Sun
TL;DR
The paper analyzes the probability p_n(F) that the n-th observation sets a Pareto record for d-dimensional i.i.d. vectors with a continuous distribution F. It introduces two dependence notions, NRPD and PRPD, via RP equivalence classes and RP ordering, and establishes that for fixed d≥2 and n≥1, p_n maps NRPD distributions onto [p^*_n,1] and PRPD distributions onto [n^{-1},p^*_n], where p^*_n is the record-setting probability for independent coordinates. The authors realize these bounds by constructing two explicit families: marginalized-Dirichlet distributions F_a (NRPD) with p_n(a) decreasing in a, and positively associated scale-mixtures widehat{F}_a (PRPD) with p_n(a) increasing in a, showing the endpoints correspond to fully independent and perfectly dependent coordinates respectively. The work provides a structured way to understand how dependence affects multivariate Pareto records and lays groundwork for extending classical univariate/multivariate records results to controlled dependent settings. Overall, it characterizes the full range of p_n achievable under NRPD and PRPD and links these behaviors to concrete distribution families, offering a path for future extensions in multivariate record theory.
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 $p_n$ 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.