Convergences of looptrees coded by excursions

TL;DR

This paper develops a new framework using excursions to encode and analyze the convergence of looptrees and introduces vernation trees, unifying trees and loops, with applications to probabilistic limit theorems.

Contribution

It introduces a novel coding of metric spaces called vernation trees from excursions, unifying trees and looptrees, and establishes convergence results with probabilistic applications.

Findings

Established a topological framework for vernation trees.

Proved limit theorems for convergences of looptrees.

Applied results to invariance principles for random discrete looptrees.

Abstract

In order to study convergences of looptrees, we construct continuum trees and looptrees from real-valued c\`adl\`ag functions without negative jumps called excursions. We then provide a toolbox to manipulate the two resulting codings of metric spaces by excursions and we formalize the principle that jumps correspond to loops and that continuous growths correspond to branches. Combining these codings creates new metric spaces from excursions that we call vernation trees. They consist of a collection of loops and trees glued along a tree structure so that they unify trees and looptrees. We also propose a topological definition for vernation trees, which yields what we argue to be the right space to study convergences of looptrees. However, those first codings lack some functional continuity, so we adjust them. We thus obtain several limit theorems. Finally, we present some probabilistic applications, such as proving an invariance principle for random discrete looptrees.