Height and contour processes of Crump-Mode-Jagers forests (I): general distribution and scaling limits in the case of short edges

TL;DR

This paper analyzes the height and contour processes of Crump-Mode-Jagers trees, revealing their distributional properties and asymptotic behavior, especially when edges are short, connecting them to Galton-Watson trees.

Contribution

It provides a novel representation of the height process in terms of ladder height processes and establishes asymptotic relations to Galton-Watson trees for short edges.

Findings

Height process distribution linked to ladder height process.

Asymptotic height process is a scaled version of Galton-Watson height process.

Contour process converges to a time-changed height process.

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 are interested in the height and contour processes encoding a general CMJ tree. We show that the one-dimensional distribution of the height process can be expressed in terms of a random transformation of the ladder height process associated with the underlying Lukasiewicz path. As an application of this result, when edges of the tree are "short" we show that, asymptotically, (1) the height process is obtained by stretching by a constant factor the height process of the associated genealogical Galton-Watson tree, (2) the contour process is obtained from the height process by a constant time change and (3) the CMJ trees converge in the sense of finite-dimensional distributions.