  Max-stable distributions and processes are important models for extreme
events and the assessment of tail risks. The full, multivariate likelihood of a
parametric max-stable distribution is complicated and only recent advances
enable its use. The asymptotic properties of the maximum likelihood estimator
in multivariate extremes are mostly unknown. In this paper we provide natural
conditions on the exponent function and the angular measure of the max-stable
distribution that ensure asymptotic normality of the estimator. We show the
effectiveness of this result by applying it to popular parametric models in
multivariate extreme value statistics and to the most commonly used families of
spatial max-stable processes.
