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

### 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 $\frac{3}{2}\ln n$, 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 $D([0,1],\r)$

Topics: Branching random walk, spinal decomposition, [ MATH.MATH-PR ] Mathematics [math]/Probability [math.PR]
Publisher: HAL CCSD
Year: 2013
OAI identifier: oai:HAL:hal-00827040v1
Provided by: Hal-Diderot

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