Coalescence on Supercritical Multi-type Branching Processes

Abstract

Let \(\mathbf {Z}_{n}=(Z_{n}^{(1)},Z_{n}^{(2)},\cdots ,Z_{n}^{(d)})\) be a d-type (d < ∞) Galton-Watson branching process. For a positive integer k ≥ 2. Pick k individuals at random from the nth generation by simple random sampling without replacement. Trace their lines of descent backward in time till they meet. Let Xn,k be the generation number of the coalescence time of these k individuals of the nth generation. We call the common ancestor of these chosen individuals in the Xn,kth generation their last common ancestor. In this paper, the limit behaviors of the distributions of Xn,k, for any integer k ≥ 2, is studied for the supercritical cases. Also, we investigate the limit distribution of joint distribution of the generation number and the type of the last common ancestor of these randomly chosen individuals and their types in the supercritical case.