  We apply a suitable modification of the functional delta method to
statistical functionals that arise from law-invariant coherent risk measures.
To this end we establish differentiability of the statistical functional in a
relaxed Hadamard sense, namely with respect to a suitably chosen norm and in
the directions of a specifically chosen "tangent space". We show that this
notion of quasi-Hadamard differentiability yields both strong laws and limit
theorems for the asymptotic distribution of the plug-in estimators. Our results
can be regarded as a contribution to the statistics and numerics of risk
measurement and as a case study for possible refinements of the functional
delta method through fine-tuning the underlying notion of differentiability
