Statistical Consistency for Risk Measures With the Lebesgue Property

When estimating the risk ρ(X) of a random variable X from historical data or Monte Carlo simulation, the asymptotic behaviour of the plug in estimator ρbn is of utmost importance. In their celebrated article [19], the Kra ̈atschmer et al. showed that any finite-valued law-invariant convex risk measure ρ defined on an Orlicz heart HΦ is statistically consistent. That is, the plug-in estimator ρbn converges in the almost sure sense to ρ(X). This result is very general, yet it does not cover the case where ρ is non-convex nor the case where ρ is defined on the entire Orlicz space L Φ. The aim of this thesis is to fill this gap. In particular, we prove that any law-invariant risk measure with the Lebesgue property is statistically consistent on the entire Orlicz space. The Lebesgue property is a continuity condition that is automatically satisfied by all convex and finite-valued risk measures on Orlicz hearts. Thus our result can be viewed as a generalization of Theorem 2.6 in [19].