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Tightness criteria for random compact sets of cadlag paths

Nic Freeman, Jan M. Swart

TL;DR

The paper extends the framework of weak convergence from single cadlag paths to random compact sets of cadlag paths under the Skorohod $J_1$ and $M_1$ topologies. It introduces a robust, crossing-based tightness criterion for random compact sets in the path space with varying time domains, and then strengthens the theory to ensure that limit points of noncrossing sets remain noncrossing. The results are then applied to the theory of weaves, showing that tightness of a single-particle motion in the classical Skorohod space drives tightness of the entire noncrossing system, with concrete examples including rescaled heavy-tailed Poisson trees and $\alpha$-stable limits. Together, the work provides a rigorous probabilistic scaffold for analyzing random compact sets of cadlag paths and their noncrossing structures, broadening the Brownian-web paradigm to jump processes.

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.

Tightness criteria for random compact sets of cadlag paths

TL;DR

The paper extends the framework of weak convergence from single cadlag paths to random compact sets of cadlag paths under the Skorohod and topologies. It introduces a robust, crossing-based tightness criterion for random compact sets in the path space with varying time domains, and then strengthens the theory to ensure that limit points of noncrossing sets remain noncrossing. The results are then applied to the theory of weaves, showing that tightness of a single-particle motion in the classical Skorohod space drives tightness of the entire noncrossing system, with concrete examples including rescaled heavy-tailed Poisson trees and -stable limits. Together, the work provides a rigorous probabilistic scaffold for analyzing random compact sets of cadlag paths and their noncrossing structures, broadening the Brownian-web paradigm to jump processes.

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.
Paper Structure (13 sections, 24 theorems, 95 equations)

This paper contains 13 sections, 24 theorems, 95 equations.

Table of Contents

  1. Main results
  2. Introduction
  3. Path space
  4. Tightness for random compact sets of paths
  5. Noncrossing sets of paths
  6. Proofs
  7. Convergence criteria
  8. Compactness criteria
  9. Tightness criteria
  10. Noncrossing paths
  11. Tightness of noncrossing sets
  12. Applications
  13. Weaves

Key Result

Theorem 1.1

Let $({\cal A}_\gamma)_{\gamma\in\Gamma}$ be a family of random variables with values in ${\cal K}_+(\Pi_{\rm c})$. Then the laws $(\mu_\gamma)_{\gamma\in\Gamma}$ with $\mu_\gamma:=\mathbb{P}[{\cal A}_\gamma\in\,\cdot\,]$ are tight with respect to the topology on $\Pi_{\rm c}$ if and only if

Theorems & Definitions (24)

  • Theorem 1.1: Tightness criterion for sets of continuous paths
  • Theorem 1.2: Tightness criteria for sets of cadlag paths
  • Theorem 1.3: Tightness criterion for noncrossing sets of paths
  • Lemma 1.4: Sets of bi-infinite paths
  • Lemma 2.1: Convergence in the Hausdorff topology
  • Lemma 2.2: The local Hausdorff topology
  • Lemma 2.3: Convergence in the J1 and M1 topologies
  • Lemma 2.4: Convergence of graphs
  • Theorem 2.5: Compactness criteria
  • Lemma 2.6: Compactness in the Hausdorff topology
  • ...and 14 more