We prove that when the distribution of a stochastic process in C[0,1]$ is in the domain of attraction of a max-stable process, then natural estimators for the extreme-value index (which is now a continuous function) and for the mean measure of the limiting Poisson process are consistent in the appropriate topologies. The ultimate goal, estimating probabilities of small (failure) sets, will be considered later.
"Weak consistency of extreme value estimators in C[0,1]." Ann. Statist. 31 (6) 1996 - 2012, December 2003. https://doi.org/10.1214/aos/1074290334