Central limit theorem for supercritical Crump-Mode-Jagers processes counted with non-individual random characteristics

TL;DR

This paper proves a central limit theorem for supercritical Crump-Mode-Jagers processes with non-independent random characteristics, extending previous work and applying it to random tree structures.

Contribution

It extends the CLT for Crump-Mode-Jagers processes by relaxing independence assumptions and applies the results to analyze fringe trees in random trees.

Findings

Established a CLT under second moment conditions.

Extended previous results to dependent characteristics.

Applied findings to study fringe trees in various random tree models.

Abstract

Consider a supercritical Crump-Mode-Jagers process $(\mathcal{Z}_{t}^{\varphi})_{t \geq 0}$ counted with a random characteristic $\varphi$ that depends on an individual's life and their descendant process up to a fixed generation. Under second moment assumptions, we establish a central limit theorem for $\mathcal{Z}_{t}^{\varphi}$ as $t \rightarrow \infty$. Our result extends the recent work of Iksanov, Kolesko, and Meiners (2024) by relaxing their assumption of independent characteristics across individuals. We further demonstrate the applicability of our results to the study of fringe trees in several important random tree families, thereby providing insights into questions raised by Holmgren and Janson (2017).