A central limit theorem for linear Kolmogorovʹs birth-growth models

A Poisson process in space-time is used to generate a linear Kolmogorovʹs birth-growth model. Points start to form on [0,L] at time zero. Each newly formed point initiates two bidirectional moving frontiers of constant speed. New points continue to form on not-yet passed over parts of [0,L]. The whole interval will eventually be passed over by the moving frontiers. Let NL be the total number of points formed. Quine and Robinson (1990) showed that if the Poisson process is homogeneous in space-time, the distribution of (NL − E[NL])√var[NL] converges weakly to the standard normal distribution. In this paper a simpler argument is presented to prove this asymptotic normality of NL for a more general class of linear Kolmogorovʹs birth-growth models.