Near-constancy of the harris function in the simple branching process

Abstract

In the theory of the simple supercritical branching process there are difficult problems concerning constancy and near-constancy phenomena. One classical case arises when the offspring generating function is polynomial of degree d Suppose the mean family size is μ and let γ be given by . Let W be the almost sure limit of the population size normalized by its expected value. Harris (1948) established that the cumulant generating function of W looks like for large s.(H is the Harris function of the title and is multiplicatively periodic, that is He reports the computation of H for the offspring generating function and finds it to be constant to 6 decimal places. Here the theory necessary for a more refined numerical study is developed and the results of that study reported. The main conclusion of the numerical study is that the Harris function is not constant (at least in this case) and that, on a suitable scale, its variation is nearly sinusoidal