  We derive the joint asymptotic distribution of empirical quantiles and
expected shortfalls under general conditions on the distribution of the
underlying observations. In particular, we do not assume that the distribution
function is differentiable at the quantile with strictly positive derivative.
Hence the rate of convergence and the asymptotic distribution for the quantile
can be non-standard, but our results show that the expected shortfall remains
asymptotically normal with a $\sqrt{n}$-rate, and we even give the joint
distribution in such non-standard cases. In the derivation we use the bivariate
scoring functions for quantile and expected shortfall as recently introduced by
Fissler and Ziegel (2016). The main technical issue is to deal with the
distinct rates for quantile and expected shortfall when applying the
argmax-continuity theorem. We also consider spectral risk measures with
finitely-supported spectral measures, and illustrate our results in a
simulation study.
