Joint Exclusivity

Abstract

We introduce joint exclusivity (JE), a relaxation of mutual exclusivity (ME) for non-negative random vectors that forbids only the simultaneous positivity of all $n$ components. Whereas ME requires the probabilities of positive outcomes to sum to at most one, a JE random vector with prescribed marginals $F_1,\dots ,F_n$ exists if and only if \[ \sum_{i=1}^n \overline{F}_i(0)\le n-1. \] An inclusion--exclusion identity shows that JE laws are determined by their $(n-1)$-dimensional margins and decomposes aggregate stop-loss premiums into those of sub-portfolios. A single wrap-around construction attains the existence bound, minimises the number of claims in convex order and, in frequency--severity models, the aggregate loss, thereby extending the safest-dependence property of ME beyond its admissible range. We also develop canonical and distortion-based constructions with freely chosen face copulas, and an $m$-exclusivity hierarchy between ME and JE with sharp two-sided stop-loss bounds.