Law of Large Numbers for Risk Measures

TL;DR

This paper proves that under certain conditions, the risk measure of sample data converges almost surely to the true risk measure, using Orlicz space theory.

Contribution

It establishes almost sure convergence of sample risk measures to the true risk measure for law invariant risk measures under integrability conditions.

Findings

Convergence holds under appropriate integrability conditions.

Results are derived using Orlicz space theory.

Provides a theoretical foundation for risk measure estimation.

Abstract

Under appropriate integrability conditions the risk measure of the sample measures for a law invariant risk measure converge almost surely to the risk measure of the sampled random variable. The results follow from general convergence theorems based on the theory of Orlicz spaces.