Computing Probabilities of Finding Extremes in a Random Set

This study examines the occurrence of extreme points in random samples of size n obtained by mapping uniformly distributed random variables through a function into a multidimensional space. We focus on the probabilities that such sets contain a unique componentwise maximum, minimum, or both. Our interest lies in the asymptotic behavior of these probabilities. We found that in some cases, for certain irregular mappings, the limits of these probabilities may fail to exist as n tends to infinity. This contrasts with our earlier work, where the assumptions of smoothness and regularity of the mapping function ensured well-behaved limits. In the present study, we investigate scenarios in which these smoothness conditions are relaxed or absent. Because the general multidimensional case is highly challenging, we restrict attention to a simpler but illustrative setting: finite random sets in the plane that lie on the graph of a real function defined over the unit interval. We present partial results in this setting and discuss open questions that remain for future research.