Forefather distribution in a variant of Galton–Watson branching process. Issue 3 (3rd July 2019)

• Record Type: Journal Article
• Authors: Laha, Arnab Kumar; Yadav, Sumit Kumar
• Is Part Of: Stochastic models. Volume 35:Issue 3(2019)
• Page Start: 269
• Page End: 283
• Publication Date: 2019-07-03
• Subjects: Age structure -- ancestors -- criticality -- discrete time multitype branching processes; Stochastic processes -- Periodicals; Probabilities -- Periodicals; 519.2
• DOI: 10.1080/15326349.2019.1578239
• Languages: English
• ISSNs: 1532-6349

Abstract

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.