On the probability of a Pareto record

Given a sequence of independent random vectors taking values in Rᵈ and having common continuous distribution function F , say that the n 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 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 ≥ 2 and n ≥ 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.