A phase transition for the probability of being a maximum among random vectors with general iid coordinates

Consider n iid real-valued random vectors of size k having iid coordinates with a general distribution function F. A vector is a maximum if and only if there is no other vector in the sample that weakly dominates it in all coordinates. Let p_k,n be the probability that the first vector is a maximum. The main result of the present paper is that if k≡k_n grows at a slower (faster) rate than a certain factor of log(n), then p_k,n→0 (resp. p_k,n→1) as n→∞. Furthermore, the factor is fully characterized as a functional of F. We also study the effect of F on p_k,n, showing that while p_k,n may be highly affected by the choice of F, the phase transition is the same for all distribution functions up to a constant factor.