Law-invariant functionals on general spaces of random variables

TL;DR

This paper extends and sharpens results on law-invariant functionals for a broad class of random variable spaces, emphasizing their structural properties and representations.

Contribution

It generalizes key results for quasiconvex, lower-semicontinuous, law-invariant functionals to larger spaces, providing a unifying framework and new insights.

Findings

Law invariance is equivalent to Schur convexity.

Law-invariant functionals are determined by their behavior on bounded variables.

The paper offers a unified perspective on quantile-based representations.

Abstract

We establish general versions of a variety of results for quasiconvex, lower-semicontinuous, and law-invariant functionals. Our results extend well-known results from the literature to a large class of spaces of random variables. We sometimes obtain sharper versions, even for the well-studied case of bounded random variables. Our approach builds on two fundamental structural results for law-invariant functionals: the equivalence of law invariance and Schur convexity, i.e., monotonicity with respect to the convex stochastic order, and the fact that a law-invariant functional is fully determined by its behaviour on bounded random variables. We show how to apply these results to provide a unifying perspective on the literature on law-invariant functionals, with special emphasis on quantile-based representations, including Kusuoka representations, dilatation monotonicity, and infimal convolutions.