A tightness criterion for fragmentations

TL;DR

This paper introduces a simple criterion for the tightness of stochastic fragmentation processes, correcting previous proofs and demonstrating applicability across various random tree models.

Contribution

It provides a new, straightforward criterion for tightness in fragmentation processes and applies it to correct and extend prior results involving random trees.

Findings

Established a tightness criterion for fragmentation processes.

Corrected the proof of a previous convergence result.

Showed applicability to multiple random tree models.

Abstract

This note presents a simple criterion for the tightness of stochastic fragmentation processes. Our work is motivated by an application to a fragmentation process derived from deleting edges in a conditioned Galton-Watson tree studied by Berzunza-Ojeda and Holmgren (2023). In that paper, while finite-dimensional convergence was established, the claimed functional convergence relied on Lemma 22 of Broutin and Marckert (2016), which is unfortunately incorrect. We show how our results can correct the proof of Berzunza-Ojeda and Holmgren (2023). Furthermore, we show the applicability of our results by establishing tightness for fragmentation processes derived from various random tree models previously studied in the literature, such as Cayley trees, trees with specified degree sequences, and $\mathbf{p}$-trees.