Defective Galton-Watson processes

TL;DR

This paper studies defective Galton-Watson processes where particles can be absorbed into a graveyard state, analyzing their long-term behavior as the process evolves and the defect probability approaches zero.

Contribution

It introduces and analyzes the properties of defective Galton-Watson processes with a focus on their asymptotic behavior as the defect probability diminishes.

Findings

Characterization of absorption probabilities

Asymptotic behavior of process trajectories

Impact of defect probability on extinction times

Abstract

The Galton-Watson process is a Markov chain modeling the population size of independently reproducing particles giving birth to $k$ offspring with probability $p_k$, $k\ge0$. In this paper we consider {\it defective} Galton-Watson processes having defective reproduction laws, so that $\sum_{k\ge0}p_k=1-\eps$ for some $\eps\in(0,1)$. In this setting, each particle may send the process to a graveyard state $\Delta$ with probability $\eps$. Such a Markov chain, having an enhanced state space $\{0,1,\ldots\}\cup\{\Delta\}$, gets eventually absorbed either at $0$ or at $\Delta$. Assuming that the process has avoided absorption until the observation time $t$, we are interested in its trajectories as $t\to\infty$ and $\eps\to0$.