  We study statistical properties of the optimal value and optimal solutions of
the Sample Average Approximation of risk averse stochastic problems. Central
Limit Theorem type results are derived for the optimal value and optimal
solutions when the stochastic program is expressed in terms of a law invariant
coherent risk measure. The obtained results are applied to hypotheses testing
problems aiming at comparing the optimal values of several risk averse convex
stochastic programs on the basis of samples of the underlying random vectors.
We also consider non-asymptotic tests based on confidence intervals on the
optimal values of the stochastic programs obtained using the Stochastic Mirror
Descent algorithm. Numerical simulations show how to use our developments to
choose among different distributions and show the superiority of the asymptotic
tests on a class of risk averse stochastic programs.
