Application to Branching Random Walk

Abstract

The purpose of this chapter is two-fold. First, we obtain a criterion for uniform integrability of intrinsic martingales $(W_{n})_{n\in \mathbb{N}_{0}}$ in the branching random walk as a corollary to Theorem 2.1.1 that provides a criterion for the a.s. finiteness of perpetuities. Second, we state a criterion for the existence of logarithmic moments of a.s. limits of $(W_{n})_{n\in \mathbb{N}_{0}}$ as a corollary to Theorems 1.3.1 and 2.1.4 While the former gives a criterion for the existence of power-like moments for suprema of perturbed random walks, the latter contains a criterion for the existence of logarithmic moments of perpetuities. To implement the task, we shall exhibit an interesting connection between these at first glance unrelated models which emerges when studying the weighted random tree associated with the branching random walk under the so-called size-biased measure.