2024/02/27 by James Allen Fill, Fill, James Allen, Ao Sun +1
Computer Science · #60G70 (Primary) 60D05 (Secondary) #Bayesian Modeling and Causal Inference #Data Structures and Algorithms (cs.DS) #FOS: Computer and information sciences #FOS: Mathematics #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.2402.17220
openalex publication_date 2024/02/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given a sequence of independent random vectors taking values in \mathbb Rd 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 pn(F) ≡ pn, 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 ≥ 2 and n ≥ 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.