Coherent distributions on the square–extreme points and asymptotics

Let C denote the family of all coherent distributions on the unit square [0,1]² , i.e. all those probability measures μ for which there exists a random vector (X,Y)~ μ , a pair (G,H) of σ -fields, and an event E such that X=P(E) , Y=P(E) almost surely. We examine the set ext(C) of extreme points of C and provide its general characterisation. Moreover, we establish several structural properties of finitely-supported elements of ext(C) . We apply these results to obtain the asymptotic sharp bound limα → ∞α·((X,Y)∈ CE|X-Y|α) = 2/e .