Coalescence on Supercritical Bellman-Harris Branching Processes

Abstract

We consider a continuous-time single-type age-dependent Bellman-Harris branching process $\{Z(t): t \geq 0\}$ with offspring distribution $\{p_j\}_{j \geq 0}$ and lifetime distribution $G$. Let $k \geq 2$ be a positive integer. If $Z(t) \geq k$, we pick $k$ individuals from those who are alive at time $t$ by simple random sampling without replacement and trace their lines of descent backward in time until they meet for the first time. Let $D_k(t)$ be the coalescence time (the death time of the most recent common ancestor) and let $X_k(t)$ be the generation number of the most recent common ancestor of these $k$ random chosen individuals. In this paper, we study the distributions of $D_k(t)$ and $X_k(t)$ and their limit distributions as $t \to \infty$.