Coherent distributions: Hilbert space approach and duality

Let X be a Bernoulli random variable with the success probability p. We are interested in tight bounds on E[f(X₁,X₂)], where Xᵢ=E[X| Fᵢ] and Fᵢ are some sigma-algebras. This problem is closely related to understanding extreme points of the set of coherent distributions. A distribution on [0,1]² is called coherent if it can be obtained as the joint distribution of (X₁, X₂) for some choice of Fᵢ. By treating random variables as vectors in a Hilbert space, we establish an upper bound for quadratic f, characterize f for which this bound is tight, and show that such f result in exposed coherent distributions with arbitrarily large support. As a corollary, we get a tight bound on cov\,(X₁,X₂) for p∈ [1/3,\,2/3]. To obtain a tight bound on cov\,(X₁,X₂) for all p, we develop an approach based on linear programming duality. Its generality is illustrated by tight bounds on E[|X₁-X₂|^α] for any α>0 and $p=1/2$.