Height and contour processes of Crump-Mode-Jagers forests (III): The binary, homogeneous universality class

TL;DR

This paper establishes conditions under which Crump-Mode-Jagers trees fall into the universality class of binary, homogeneous CMJ trees, expanding understanding of their scaling limits and classification.

Contribution

It identifies general conditions for CMJ trees to belong to the binary, homogeneous universality class, extending previous classifications of CMJ trees.

Findings

Renewal processes satisfy the identified conditions.

Conditions prevent accumulation of atoms near the origin.

Results generalize the universality classification of CMJ trees.

Abstract

This paper belongs to a series of papers aiming to investigate scaling limits of Crump-Mode-Jagers (CMJ) trees. In the previous two papers we identified general conditions under which CMJ trees belong to the universality class of Galton-Watson and Bellman-Harris processes. In this paper we identify general conditions for CMJ trees to belong to the universality class of binary, homogeneous CMJ trees. These conditions state that the offspring process should 'look like' a renewal process, and also that it should not accumulate too many atoms near the origin. We show in particular that any renewal process satisfies these conditions.