On the probability of a Pareto record

Given a sequence of independent random vectors taking values in ℝ^d and having common continuous distribution F, say that the n^ th observation sets a (Pareto) record if it is not dominated (in every coordinate) by any preceding observation. Let p_n(F) ≡ p_n, 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 (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 p_n 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.