  We present a simple unifying treatment of a broad class of applications from
statistical mechanics, econometrics, mathematical finance, and insurance
mathematics, where (possibly subordinated) L\'evy noise arises as a scaling
limit of some form of continuous-time random walk (CTRW). For each application,
it is natural to rely on weak convergence results for stochastic integrals on
Skorokhod space in Skorokhod's J1 or M1 topologies. As compared to earlier and
entirely separate works, we are able to give a more streamlined account while
also allowing for greater generality and providing important new insights. For
each application, we first elucidate how the fundamental conclusions for J1
convergent CTRWs emerge as special cases of the same general principles, and we
then illustrate how the specific settings give rise to different results for
strictly M1 convergent CTRWs.
