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arXiv · 1906.03945

Coalescence for supercritical Galton-Watson processes with immigration

Hua-Ming Wang·Lulu Li·Huizi Yao
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Abstract

In this paper, we consider Galton-Watson processes with immigration. Pick $i(\ge2)$ individuals randomly without replacement from the $n$-th generation and trace their lines of descent back in time till they coalesce into $1$ individual in a certain generation, which we denote by $X_{i,1}^n$ and is called the coalescence time. Firstly, we give the probability distribution of $X_{i,1}^n$ in terms of the probability generating functions of both the offspring distribution and the immigration law. Then by studying the limit behaviors of various functionals of the Galton-Watson process with immigration, we find the limit distribution of $X_{2,1}^n$ as $n\rightarrow\infty.$

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Hua-Ming Wang, Lulu Li, Huizi Yao. 2019-06-10. Coalescence for supercritical Galton-Watson processes with immigration. https://arxiv.org/abs/1906.03945

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