The central limit theorem for the supercritical branching random walk, and related results

Abstract

In a branching random walk each family arrives equipped with its members' positions relative to their parent, these being i.i.d. copies of some point process X. The supercritical case is considered so the mean family size m > 1, and X has intensity measure mν, where ν is a probability measure. The nth generation of the process, Z ( n) , then has intensity measure m nν n∗ so it is natural to expect m - n Z ( n) to exhibit ‘central limit’ behaviour. Such a result, corresponding results for stable laws, their local analogues and some similar results when a Seneta-Heyde norming is needed are all obtained here. The main result, from which these are derived, provides a good approximation to the ‘characteristic function’ of Z ( n) for the generation dependent version of the process.