Cantelli's bounds for generalized tail inequalities in Euclidean spaces

TL;DR

This paper extends Cantelli's inequalities to generalized tail probabilities in Euclidean spaces using cone duality, providing sharp bounds for probabilities involving vector inequalities defined by cones.

Contribution

It introduces sharp Cantelli-type bounds for tail probabilities of random vectors with respect to cone-induced partial orders, generalizing classical scalar inequalities.

Findings

Derived sharp bounds for generalized tail probabilities

Utilized cone duality to scalarize vector inequalities

Minimized classical Cantelli bounds over scalarized inequalities

Abstract

Let $X$ be a centered random vector in a finite dimensional real inner product space $\mathcal{E}$. For a subset $C$ of the ambient vector space $V$ of $\mathcal{E}$ and $x,\,y\in V$, write $x\preceq_C y$ if $y-x\in C$. When $C$ is a closed convex cone in $\mathcal{E}$, then $\preceq_C$ is a pre-order on $V$, whereas if $C$ is a proper cone in $\mathcal{E}$, then $\preceq_C$ is actually a partial order on $V$. In this paper we give sharp Cantelli's type inequalities for generalized tail probabilities like $\text{Pr}\{X\succeq_C b\}$ for $b\in V$. These inequalities are obtained by ``scalarizing'' $X\succeq_C b$ via cone duality and then by minimizing the classical univariate Cantelli's bound over the scalarized inequalities.