Tightness criteria for random compact sets of cadlag paths
Pith reviewed 2026-05-23 06:14 UTC · model grok-4.3
The pith
Tightness criteria for random compact sets of cadlag paths are given in the Skorohod J1 and M1 topologies.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
Tightness criteria are stated for random compact sets of cadlag paths under both the Skorohod J1 and M1 topologies on the space of such sets. The M1 version is constructed so that any weak limit of a non-crossing system is again non-crossing. The criteria are applied to obtain limit theorems for rescaled heavy-tailed Poisson trees and for weaves.
What carries the argument
Tightness criteria expressed in the Skorohod J1 and M1 topologies on the space of compact sets of cadlag paths; the criteria generalize the continuous-path case and, in the M1 topology, automatically preserve non-crossing.
If this is right
- Any sequence of random compact sets satisfying the J1 or M1 criteria possesses at least one weakly convergent subsequence.
- Under the M1 criteria, the weak limit of a non-crossing collection remains non-crossing.
- The criteria apply directly to suitably rescaled heavy-tailed Poisson trees.
- The criteria apply to the general construction of weaves.
Where Pith is reading between the lines
- The same criteria could be checked in other models that produce random collections of cadlag paths, such as certain queueing or risk processes.
- Numerical simulation of finite approximations to Poisson trees could be used to test whether the non-crossing property is preserved in practice under the M1 topology.
- The extension from continuous to cadlag paths suggests that analogous criteria might exist for compact sets valued in other Skorohod-type spaces, such as those with jumps of bounded variation.
Load-bearing premise
The Skorohod topologies on the space of compact sets of cadlag paths admit a direct generalization of the tightness criteria already known for continuous paths.
What would settle it
A concrete sequence of random compact sets of cadlag paths that satisfies the stated moment and modulus-of-continuity conditions yet fails to be tight in the Skorohod J1 or M1 topology on the space of compact sets.
read the original 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.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper provides tightness criteria for random variables taking values in the space of compact sets of cadlag real-valued paths, formulated in both the Skorohod J1 and M1 topologies. This extends prior results that were restricted to continuous paths and motivated by the Brownian web. For the M1 topology, the criteria are extended to ensure that non-crossing systems of paths have weak limit points that remain non-crossing; the result is illustrated via a rescaling of heavy-tailed Poisson trees and a general application to weaves.
Significance. If the derivations hold, the work supplies useful extensions of tightness tools to discontinuous path systems, broadening applicability beyond continuous settings in stochastic processes and random media. The non-crossing preservation property under M1 is a concrete strength that supports structural control in limits, and the examples provide direct applicability checks.
minor comments (3)
- [Abstract / §1] The abstract and introduction would benefit from an explicit statement of the precise form of the tightness criteria (e.g., moment or modulus-of-continuity conditions) rather than a high-level description.
- [§2] Notation for the space of compact subsets (e.g., the precise definition of the metric inducing the topology on compact sets) should be introduced before the statement of the main theorems to improve readability.
- [§4] In the M1 non-crossing result, clarify whether the preservation holds for all limit points or only for subsequences; a brief remark on the role of the non-crossing assumption in the proof would help.
Simulated Author's Rebuttal
We thank the referee for their positive summary of the manuscript, the assessment of its significance, and the recommendation of minor revision. No specific major comments were raised in the report.
Circularity Check
No significant circularity
full rationale
The paper claims to extend tightness criteria from continuous-path cases (motivated by Brownian web) to cadlag paths under Skorohod J1/M1 topologies, with an additional non-crossing preservation result for M1 exemplified on Poisson trees and weaves. No quoted equations, definitions, or self-citations reduce any central claim to its own inputs by construction; the criteria are framed as direct generalizations with independent topological grounding. The derivation chain is self-contained against external benchmarks and prior literature without load-bearing self-referential steps.
discussion (0)
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