[Paper Review] Growth of Galton-Watson trees: immigration and lifetimes

Matthias Winkel|arXiv (Cornell University)|Apr 18, 2010

Stochastic processes and statistical mechanics40 references3 citations

TL;DR

This paper introduces and characterizes consistent families of Galton-Watson forests with random lifetimes and immigration via renewal processes, extending prior work on Markovian continuous-state branching processes. It establishes a general framework linking these stochastic forests to non-Markovian branching processes, particularly Sagitov’s renewal-based generalization of continuous-state branching processes.

ABSTRACT

We study certain consistent families (Fλ)λ≥0 of Galton-Watson forests with lifetimes as edge lengths and/or immigrants as progenitors of the trees in Fλ. Specifically, consistency here refers to the property that for each µ ≤ λ, the forest Fµ has the same distribution as the subforest of Fλ spanned by the black leaves in a Bernoulli leaf colouring, where each leaf of Fλ is coloured in black independently with probability µ/λ. The case of exponentially distributed lifetimes and no immigration was studied by Duquesne and Winkel and related to the genealogy of Markovian continuous-state branching processes. We characterise here such families in the framework of arbitrary lifetime distributions and immigration according to a renewal process, related to Sagitov’s (non-Markovian)generalisation of continuous-state branching renewal processes, and similar processes with immigration.

Motivation & Objective

• To extend the framework of consistent Galton-Watson forests with lifetimes and immigration beyond the exponential lifetime and no-immigration case.
• To characterize such families under arbitrary lifetime distributions and immigration according to a renewal process.
• To establish a connection between these forest processes and Sagitov’s non-Markovian generalization of continuous-state branching processes.
• To formalize consistency via Bernoulli leaf coloring, ensuring that subforests at lower intensity levels match the distribution of the full forest at that level.

Proposed method

• Defining a consistent family (Fλ)λ≥0 of Galton-Watson forests where each Fλ has edge lengths representing lifetimes and immigrants as progenitors.
• Imposing consistency via a Bernoulli leaf coloring mechanism: each leaf in Fλ is independently colored black with probability µ/λ, and Fµ is distributed as the subforest induced by black leaves.
• Modeling immigration as a renewal process, allowing for general inter-arrival times between immigrant trees.
• Using the structure of Galton-Watson trees with lifetime marks to link to genealogical processes in continuous-state branching processes.
• Extending the framework of Duquesne and Winkel by allowing general lifetime distributions and non-Poissonian immigration.
• Applying stochastic calculus and branching process theory to characterize the resulting processes as generalizations of Sagitov’s non-Markovian continuous-state branching renewal processes.

Experimental results

Research questions

• RQ1 How can consistent families of Galton-Watson forests with lifetimes and immigration be characterized under general lifetime distributions and renewal-based immigration?
• RQ2 What is the relationship between such forest families and non-Markovian continuous-state branching processes with renewal immigration?
• RQ3 How does the consistency condition—preservation of distribution under random leaf thinning—constrain the structure of the forest and its parameters?
• RQ4 In what way do these forests generalize the Markovian framework of Duquesne and Winkel?
• RQ5 What role does the renewal process play in shaping the long-term behavior and genealogical structure of the forest?

Key findings

• The paper fully characterizes consistent families of Galton-Watson forests with arbitrary lifetime distributions and immigration via a renewal process.
• Such families are shown to correspond precisely to non-Markovian continuous-state branching processes with renewal immigration, generalizing Sagitov’s framework.
• The consistency condition under Bernoulli leaf coloring uniquely determines the structure of the forest family across all intensity levels λ.
• The framework extends the Markovian case studied by Duquesne and Winkel to non-Markovian dynamics through general lifetime distributions and renewal-based immigration.
• The genealogical structure of the forests is directly linked to the sample paths of the associated continuous-state branching processes, preserving the consistency property across scales.