Height and contour processes of Crump-Mode-Jagers forests (II): The Bellman-Harris universality class

TL;DR

This paper identifies a condition under which the height and contour processes of Crump-Mode-Jagers forests fall into the Bellman-Harris universality class, revealing new insights into their asymptotic independence and broad applicability.

Contribution

It introduces a simple condition linking CMJ forests to Bellman-Harris processes, applicable to many CMJ models including those with finite variance offspring distributions.

Findings

Height and contour processes belong to Bellman-Harris universality class under the condition.

Condition formalizes asymptotic independence between chronological and genealogical structures.

Proves a general tightness result for these processes.

Abstract

Crump-Mode-Jagers (CMJ) trees generalize Galton-Watson trees by allowing individuals to live for an arbitrary duration and give birth at arbitrary times during their life-time. In this paper, we exhibit a simple condition under which the height and contour processes of CMJ forests belong to the universality class of Bellman-Harris processes. This condition formalizes an asymptotic independence between the chronological and genealogical structures. We show that it is satisfied by a large class of CMJ processes and in particular, quite surprisingly, by CMJ processes with a finite variance offspring distribution. Along the way, we prove a general tightness result.