An Aldous-type tightness characterisation for the $M_1$ topology: replacing global modulus control by local control at stopping times

Abstract

Aldous' criterion provides a stopping-time characterisation of tightness in Skorokhod's $J_1$ topology and is one of the standard tools for proving functional limit theorems. For the weaker $M_1$ topology, where limiting jumps can be approximated by clustered jumps or steep continuous motion, no analogous characterisation has been available and tightness classically requires controlling a global oscillation modulus over all nearby triples of times. We prove such an Aldous-type characterisation of $M_1$ tightness, replacing the global modulus condition by local estimates at stopping times ruling out rapid reversals after a large excursion. Together with the standard compact containment and boundary conditions, these estimates are both necessary and sufficient for $M_1$ tightness. As a consequence, we derive tractable and general sufficient conditions for additive functionals of strong Markov processes. They reduce the verification of $M_1$ tightness to short-time estimates for the processes restarted from its state at suitable stopping times.