Coalescence in Bellman-Harris and multi-type branching processes.

摘要

For branching processes, there are many well-known limit theorems regarding the evolution of the population in the future time. In this dissertation, we investigate the other direction of the evolution, that is, the past of the processes. We pick some individuals at random by simple random sampling without replacement and trace their lines of descent backward in time until they meet. We study the coalescence problem of the discrete-time multi-type Galton-Watson branching process and both the continuous-time single-type and multi-type Bellman-Harris branching processes including the generation number, the death time (in the continuous-time processes) and the type (in the multi-type processes) of the last common ancestor (also called the most recent common ancestor) of the randomly chosen individuals for the different cases (supercritical, critical, subcritical and explosive).