On the probability of a Pareto record

Given a sequence of independent random vectors taking values in Rᵈ and having common continuous distribution F, say that the nᵗʰ 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 nᵗʰ 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 (NRSPD) and positive record-setting probability dependence (PRSPD), 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 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.