  We consider regenerative processes with values in some Polish space. We
define their \epsilon-big excursions as excursions e such that f(e)>\epsilon,
where f is some given functional on the space of excursions which can be
thought of as, e.g., the length or the height of e. We establish a general
condition that guarantees the convergence of a sequence of regenerative
processes involving the convergence of \epsilon-big excursions and of their
endpoints, for all \epsilon in a countable set whose closure contains 0.
Finally, we provide various sufficient conditions on the excursion measures of
this sequence for this general condition to hold and discuss possible
generalizations of our approach to processes that can be written as the
concatenation of i.i.d. paths.
