Forefather distribution in a variant of Galton–Watson branching process

Abstract

In this paper, we consider a variant of a discrete time Galton–Watson Branching Process in which an individual is allowed to survive for more than one (but finite) number of generations and may also give birth to offsprings more than once. We model the process using multitype branching process and derive conditions on the mean matrix that determines the long-run behavior of the process. Next, we analyze the distribution of the number of forefathers in a given generation. Here, number of forefathers of an individual is defined as all the individuals since zeroth generation who have contributed to the birth of the individual under consideration. We derive an exact expression for expected number of individuals in a given generation having a specified number of forefathers. Using this exact expression, we provide a detailed analysis for a simple illustrative case. Some interesting insights and possible applications are also discussed.