  Block maxima methods constitute a fundamental part of the statistical toolbox
in extreme value analysis. However, most of the corresponding theory is derived
under the simplifying assumption that block maxima are independent observations
from a genuine extreme value distribution. In practice however, block sizes are
finite and observations from different blocks are dependent. Theory respecting
the latter complications is not well developed, and, in the multivariate case,
has only recently been established for disjoint blocks of a single block size.
We show that using overlapping blocks instead of disjoint blocks leads to a
uniform improvement in the asymptotic variance of the multivariate empirical
distribution function of rescaled block maxima and any smooth functionals
thereof (such as the empirical copula), without any sacrifice in the asymptotic
bias. We further derive functional central limit theorems for multivariate
empirical distribution functions and empirical copulas that are uniform in the
block size parameter, which seems to be the first result of this kind for
estimators based on block maxima in general. The theory allows for various
aggregation schemes over multiple block sizes, leading to substantial
improvements over the single block length case and opens the door to further
methodology developments. In particular, we consider bias correction procedures
that can improve the convergence rates of extreme-value estimators and shed
some new light on estimation of the second-order parameter when the main
purpose is bias correction.
