Scaling limit of the path leading to the leftmost particle in a branching random walk

2013/05/29 by Xinxin Chen, Chen, Xinxin
Economics, Econometrics and Finance · Mathematics · #Markov Chains and Monte Carlo Methods #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #math.PR

paper · pdf · doi:10.48550/arxiv.1305.6723

arxiv created 2013/05/29 · arxiv updated 2013/05/30

Abstract

We consider a discrete-time branching random walk defined on the real line, which is assumed to be supercritical and in the boundary case. It is known that its leftmost position of the n-th generation behaves asymptotically like (3)/(2)ln n, provided the non-extinction of the system. The main goal of this paper, is to prove that the path from the root to the leftmost particle, after a suitable normalizatoin, converges weakly to a Brownian excursion in D([0,1],\r).

Related

• On the Maximal Displacement of Near-critical Branching Random Walks
• On the maximal displacement of critical branching random walk in random environment
• The near-critical Gibbs measure of the branching random walk
• Convergence in law for the branching random walk seen from its tip
• Fine asymptotics for the consistent maximal displacement of branching Brownian motion