Asymptotic normality of processes with a branching structure

Abstract

We define a stochastic process in terms of cumulative sums of the sequence of integer-valued random variables in such a way that has a branching structure; in particular when are iid and non-negativeis is a Bienayme-Galton-Watson branching process. We establish distributional limits involving the standard normal distribution for Xn and for , with a variety of standardizations and with various conditioning events. The interplay between the asymptotic behaviour of Xn and that of Tn is central to our investigations