Studies in one dimensional branching random walks.

Abstract

This thesis deals with three problems arising from branching random walks. The first problem studies the leftmost path (compared with the leftmost particle) of branching random walks. Let T denote a rooted b-ary tree and let {Sv}vET denote a branching random walk indexed by the vertices of the tree, where the increments are i.i.d. and possess a logarithmic moment generating function A(.). Let mn denote the minimum of the variables Sv over all vertices at the nth generation, denoted by Dn. Under mild conditions, mn/n converges almost surely to a constant, which for convenience may be taken to be 0. With Sv = max{Sw : w is on the geodesic connecting the root to v} define Ln = minv2Dn Sv. We prove that Ln=n/n1/3 converges almost surely to an explicit constant lo. The second problem studies the tightness of maxima (the displacement of the rightmost particle) of generalized branching random walks on the real line R that allow time dependence and local dependence between siblings. At time n, Fn(.) is used to denote the distribution function of the maximum. Under appropriate tail assumptions on the branching laws and offspring displacement distributions, we prove that Fn(.-Med(Fn)) is tight in n. The main part of the argument is to demonstrate the exponential decay of the right tail 1 - Fn(.-Med(Fn)). The third problem studies the maximum of branching random walks in a class of time inhomogeneous environments. Specifically, binary branching random walks with Gaussian increments will be considered, where the variances of the increments change over time macroscopically. We find the asymptotics of the maximum up to an OP (1) (stochastically bounded) error, and focus on the following phenomena: the profile of the variance matters, both to the leading (velocity) term and to the logarithmic correction term, and the latter exhibits a phase transition.