  In the past decades, weak convergence theory for stochastic processes has
become a standard tool for analyzing the asymptotic properties of various
statistics. Routinely, weak convergence is considered in the space of bounded
functions equipped with the supremum metric. However, there are cases when weak
convergence in those spaces fails to hold. Examples include empirical copula
and tail dependence processes and residual empirical processes in linear
regression models in case the underlying distributions lack a certain degree of
smoothness. To resolve the issue, a new metric for locally bounded functions is
introduced and the corresponding weak convergence theory is developed.
Convergence with respect to the new metric is related to epi- and
hypo-convergence and is weaker than uniform convergence. Still, for continuous
limits, it is equivalent to locally uniform convergence, whereas under mild
side conditions, it implies $L^p$ convergence. For the examples mentioned
above, weak convergence with respect to the new metric is established in
situations where it does not occur with respect to the supremum distance. The
results are applied to obtain asymptotic properties of resampling procedures
and goodness-of-fit tests.
