  Extreme-value theory for random vectors and stochastic processes with
continuous trajectories is usually formulated for random objects all of whose
univariate marginal distributions are identical. In the spirit of Sklar's
theorem from copula theory, such marginal standardization is carried out by the
pointwise probability integral transform. Certain situations, however, call for
stochastic models whose trajectories are not continuous but merely upper
semicontinuous (usc). Unfortunately, the pointwise application of the
probability integral transform to a usc process does in general not preserve
the upper semicontinuity of the trajectories. In the present work, we give
sufficient conditions for marginal standardization of usc processes to be
possible, and we state a partial extension of Sklar's theorem for usc
processes. We specialize the results to max-stable processes whose marginal
distributions and normalizing sequences are allowed to vary with the
coordinate.
