Discriminated Emergence of Weibull and Log-normal Distributions in Branching Processes

摘要: We consider the Yule-type multiplicative growth and division process, and describe ubiquitous emergence of Weibull and log-normal distributions in a single framework. With the help of the integral transform and series expansion, we show that both distributions serve as asymptotic solutions of the time evolution equation for the branching process. In particular, the maximum likelihood method is employed to discriminate emergence of the Weibull and the log-normal distributions. Further, the detailed conditions for the distinguished emergence of the Weibull distribution are probed. It is observed that emergence of the two different types of distribution depends on the manner of the division process. Numerical simulations are also carried out, confirming those results obtained analytically.