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An Aldous-type tightness characterisation for the M1 topology: replacing global modulus control by local control at stopping times

  • Grégoire Szymanski
  • Fabrice Wunderlich
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Abstract

Aldous' criterion provides a stopping-time characterisation of tightness in Skorokhod's J1 topology and is one of the standard tools for proving functional limit theorems. For the weaker M1 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 M1 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 M1 tightness. As a consequence, we derive tractable and general sufficient conditions for additive functionals of strong Markov processes. They reduce the verification of M1 tightness to short-time estimates for the processes restarted from its state at suitable stopping times.

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Cite

BibTeXRIS
BibTeX
@misc{szymanski2026aldoustype,
  title = {{An Aldous-type tightness characterisation for the $M_1$ topology: replacing global modulus control by local control at stopping times}},
  author = {Grégoire Szymanski and Fabrice Wunderlich},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.27900},
  eprint = {2610.27900},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.27900},
}
RIS
TY  - UNPB
TI  - An Aldous-type tightness characterisation for the M_1 topology: replacing global modulus control by local control at stopping times
AU  - Grégoire Szymanski
AU  - Fabrice Wunderlich
PY  - 2026
DA  - 2026/10/20
PB  - arXiv
DO  - 10.48550/arxiv.2610.27900
UR  - https://arxiv.org/abs/2610.27900
AB  - 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.
ER  -