A note on continuity and asymptotic consistency of measures of risk and variability

TL;DR

This paper proves that certain risk and variability measures are automatically continuous and consistent under specific mathematical conditions, improving previous results and extending their applicability.

Contribution

It demonstrates that convex, order-bounded functionals on Frechet lattices are norm continuous and that order-continuous, law-invariant functionals on Orlicz spaces are strongly consistent, extending prior findings.

Findings

Convex, order-bounded functionals on Frechet lattices are norm continuous.

Order-continuous, law-invariant functionals on Orlicz spaces are strongly consistent.

Results apply to many deviation and variability measures.

Abstract

In this short note, we show that every convex, order bounded above functional on a Frechet lattice is automatically norm continuous. This improves a result in \cite{RS06} and applies to many deviation and variability measures. We also show that an order-continuous, law-invariant functional on an Orlicz space is strongly consistent everywhere, extending a result in \cite{KSZ14}.