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