A note on stochastic dominance and compactness

TL;DR

This paper explores the lattice structures induced by stochastic dominance orders, demonstrating their completeness properties and how certain bounds and topologies relate to compactness and approximation of measures.

Contribution

It establishes that stochastic dominance orders induce Dedekind super complete lattices and characterizes compactness of sublattices via weak and Wasserstein-$1$ topologies.

Findings

Stochastic dominance induces Dedekind super complete lattices.

Suprema and infima can be approximated by sequences in specific topologies.

Compactness of sublattices relates to their completeness under stochastic dominance.

Abstract

In this work, we discuss completeness for the lattice orders of first and second order stochastic dominance. The main results state that, both, first and second order stochastic dominance induce Dedekind super complete lattices, i.e.~lattices in which every bounded nonempty subset has a countable subset with identical least upper bound and greatest lower bound. Moreover, we show that, if a suitably bounded set of probability measures is directed (e.g.~a lattice), then the supremum and infimum w.r.t.~first or second order stochastic dominance can be approximated by sequences in the weak topology or in the Wasserstein-$1$ topology, respectively. As a consequence, we are able to prove that a sublattice of probability measures is complete w.r.t.~first order stochastic dominance or second order stochastic dominance and increasing convex order if and only if it is compact in the weak topology or in the Wasserstein-$1$ topology, respectively. This complements a set of characterizations of tightness and uniform integrability, which are discussed in a preliminary section.