Cantelli’s Bounds for Generalized Tail Inequalities

Let X be a centered random vector in a finite-dimensional real inner product space E. For a subset C of the ambient vector space V of E and x,y∈V, write x⪯Cy if y−x∈C. If C is a closed convex cone in E, then ⪯C is a preorder on V, whereas if C is a proper cone in E, then ⪯C is actually a partial order on V. In this paper, we give sharp Cantelli-type inequalities for generalized tail probabilities such as PrX⪰Cb for b∈V. These inequalities are obtained by “scalarizing” X⪰Cb via cone duality and then by minimizing the classical univariate Cantelli’s bound over the scalarized inequalities. Three diverse applications to random matrices, tails of linear images of random vectors, and network homophily are also given.