Coalescence in critical and subcritical Galton-Watson branching processes

Abstract

In a Galton-Watson branching process that is not extinct by the nth generation and has at least two individuals, pick two individuals at random by simple random sampling without replacement. Trace their lines of descent back in time till they meet. Call that generation X_n a pairwise coalescence time. Similarly, let Y_n denote the coalescence time for the whole population of the nth generation conditioned on the event that it is not extinct. In this paper the distributions of X_n and Y_n, and their limit behaviors as n → ∞ are discussed for both the critical and subcritical cases.