COALESCENCE IN SUBCRITICAL BELLMAN-HARRIS AGE-DEPENDENT BRANCHING PROCESSES

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 -> infinity.