Mass erasure on infinitely measured $\mathbb{R}$-trees I: construction and limit theorems

Abstract

This paper is the first in a two-part series, seeking to i) extend the theory of mass erasure developed by Duquesne and Winkel to a setting with infinite measures, and ii) apply this theory to obtain invariance principles for Galton--Watson and Lévy forests in the most general setting where the trees may be supercritical and not satisfy Grey's condition. We consider $\mathbb{R}$-trees equipped with a suitable subclass of boundedly finite measures, which we call discretely infinite measures. Let $(T, d, ρ)$ be a complete and separable $\mathbb{R}$-tree, let $μ$ be a discretely infinite measure on $T$, and let $h > 0$. The $h$-mass-erased subtree is obtained by removing all fringe subtrees with $μ$-mass strictly less than $h$, and we prove that all $h$-mass-erased subtrees have a discrete branching structure. The mass-erased subtrees may be equipped with associated measures, which gives rise to a family of operators $(\mathscr{E}_h)_{h \geq 0}$ which forms a semigroup on the space of discretely infinite measures. The operators may be lifted to an appropriate space of Gromov-vague isometry classes. Given a sequence $\boldsymbolμ_n = [T_n, d_n, ρ_n, μ_n]$, $n \in \mathbb{N}$, of suitable isometry classes, we say that they converge vaguely in the sense of mass erasure if their mass erasures $(\mathscr{E}_h \boldsymbolμ_n)_{n \in \mathbb{N}}$ converge in the Gromov-vague topology for all $h > 0$. We prove that by imposing suitable tightness conditions on $(\boldsymbolμ_n)_{n \in \mathbb{N}}$, Gromov-vague convergence implies vague convergence in the sense of mass erasure. Under appropriate tightness, we identify conditions which relate the two notions of convergence, and prove that either mode of convergence implies local Gromov--Hausdorff convergence of the mass-erased subtrees. The applications to Galton--Watson and Lévy forests are considered in part two.