Tightness criteria for random compact sets of cadlag paths

TL;DR

This paper establishes tightness criteria for random compact sets of cadlag paths under Skorohod topologies, extending previous work to include non-continuous paths and applications like Poisson trees.

Contribution

It introduces new tightness criteria for cadlag path sets in Skorohod topologies, including non-crossing systems and their weak limits, with applications to Poisson trees and weaves.

Findings

Criteria for tightness in Skorohod J1 and M1 topologies.

Extension of tightness criteria to non-crossing path systems.

Application to heavy-tailed Poisson trees and weaves.

Abstract

We give tightness criteria for random variables taking values in the space of all compact sets of cadlag real-valued paths, in terms of both the Skorohod J1 and M1 topologies. This extends earlier work motivated by the study of the Brownian web that was concerned only with continuous paths. In the M1 case, we give a natural extension of our tightness criteria which ensures that non-crossing systems of paths have weak limit points that are also non-crossing. This last result is exemplified through a rescaling of heavy tailed Poisson trees and a more general application to weaves.