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Fifth Colloquium on Mathematics and Computer Science DMTCS proc. AI, 2008, 421436 A Note on the Transience of Critical

Summary: Fifth Colloquium on Mathematics and Computer Science DMTCS proc. AI, 2008, 421­436
A Note on the Transience of Critical
Branching Random Walks on the Line
Gerold Alsmeyer and Matthias Meiners
Institut f¨ur Mathematische Statistik, Fachbereich Mathematik, Einsteinstraße 62, D-48149 M¨unster, Germany
Gantert and M¨uller (2006) proved that a critical branching random walk (BRW) on the integer lattice is transient by
analyzing this problem within the more general framework of branching Markov chains and making use of Lyapunov
functions. The main purpose of this note is to show how the same result can be derived quite elegantly and even
extended to the nonlattice case within the theory of weighted branching processes. This is done by an analysis of
certain associated random weighted location measures which, upon taking expectations, provide a useful connection
to the well established theory of ordinary random walks with i.i.d. increments. A brief discussion of the asymptotic
behavior of the left- and rightmost particles in a critical BRW as time goes to infinity is provided in the final section
by drawing on recent work by Hu and Shi (2008).
Keywords: branching random walk, critical regime, recurrence, transience, minimal and maximal position, random
weighted location measure, renewal theory
1 Introduction
Consider a cloud of particles which moves on the line as follows. Initially there is one particle sitting at
the origin which after one unit of time splits into a random number of new particles having distribution
(pj)j0, where p0 = 0. The daughter particles are then independently displaced relative to their mother's
site in accordance with the same step size distribution Q, say. This process continues indefinitely, i.e., each


Source: Alsmeyer, Gerold - Institut für Mathematische Statistik, Westfälische Wilhelms-Universität Münster


Collections: Mathematics