Branching Random Walks Conditioned on Particle Numbers

In this paper, we consider a pruned Galton-Watson tree conditioned to have k particles in generation n, i.e. we take a Gallon-Watson tree satisfying Z(n) = k, and delete all branches that die before generation n. We show that with k fixed and n -> infinity, the first n generations of this tree can be described by an explicit probability measure P-k(st). As an application, we study a branching random walk (V-u)(u is an element of T) indexed by such a pruned Galton-Watson tree T, and give the asymptotic tail behavior of the span and gap statistics of its k particles in generation n, (V-u)(vertical bar u vertical bar=n). This is the discrete version of Ramola et al. (Chaos Solitons Fractals 74:79-88, 2015), generalized to arbitrary offspring and displacement distributions with moment constraints.