Bienayme Galton Watson Branching Process

Sivaprasad Madhira and Shailaja Deshmukh ()
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Sivaprasad Madhira: Savitribai Phule Pune University
Shailaja Deshmukh: Savitribai Phule Pune University

Chapter Chapter 5 in Introduction to Stochastic Processes Using R, 2023, pp 273-320 from Springer

Abstract: Abstract Branching processes are Markov chains with countably infinite state space. While there are many variations of branching processes, in this chapter, only Bienayme-Galton-Watson (BGW) branching process is considered. Branching processes model the evolution of populations over generations. An important problem is to obtain the probability of ultimate extinction of the population represented by the branching process. In Sect. 1, a brief history of the process is given. In Sect. 2, it is shown that a BGW branching process is a non-ergodic Markov chain on the set of non-negative integers. In Sect. 3, a recurrence relation satisfied by the probability generating function of population size in the nth generation is derived. The fundamental theorem of the branching process that relates the probability of extinction of the process to the mean of the off-spring distribution is proved in Sect. 4. The computation of extinction probability graphically and algebraically is considered in Sect. 5. Finally, Sect. 6 contains the relevant R codes that are used in solving various examples.

Date: 2023
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DOI: 10.1007/978-981-99-5601-2_5

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