  We present a general framework for weak convergence to decorated L\'evy
processes in enriched spaces of c\`adl\`ag functions for vector-valued
processes arising in deterministic systems. Applications include uniformly
expanding maps and unbounded observables as well as nonuniformly
expanding/hyperbolic maps with bounded observables. The latter includes
intermittent maps and dispersing billiards with flat cusps. In many of these
examples, convergence fails in all of the Skorohod topologies. Moreover, the
enriched space picks up details of excursions that are not recorded by Skorohod
or Whitt topologies.
