Coalescence in subcritical Bellman-Harris age-dependent branching processes

Abstract

We consider a continuous-time, single-type, age-dependent Bellman-Harris branching process. We investigate the limit distribution of the point process A(t)={a_t,i: 1≤ i≤ Z(t)}, where a_t,i is the age of the ith individual alive at time t, 1≤ i≤ Z(t), and Z(t) is the population size of individuals alive at time t. Also, if Z(t)≥ k, k≥2, is a positive integer, 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 last common ancestor) of these k random chosen individuals. We study the distribution of D_k(t) and its limit distribution as t→∞.