Asymptotic Normality of Processes with a Branching Structure

We define a stochastic process fX n ; n = 0; 1; 2; : : :g in terms of cumulative sums of the sequence K 1 ; K 2 ; : : : of integer-valued random variables in such a way that fX n g has a branching structure; in particular when fK i g are iid and non-negative, fX n g is a Bienaym'e-Galton-Watson branching process. We establish distributional limits involving the standard normal distribution for X n and for T n := P n j=0 X j , with a variety of standardizations and with various conditioning events. The interplay between the asymptotic behaviour of X n and that of T n is central to our investigations. Keywords: Asymptotic normality, Bienaym'e-Galton-Watson branching process, Wiener process, functional central limit theorem. 1 1 INTRODUCTION 1.1 THE MODEL AND THE PROBLEM Let K 1 ; K 2 ; : : : be integer-valued random variables (rv's) defined on the same probability space(\Omega ; F ; P) and define sequences fX n ; n = 0; 1; 2; : : :g and fT n ; n = 0; 1; 2; : : :g by X 0 = 1, X 1 ..