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

Xinxin Chen (1)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

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 $D([0,1],\r)$.

Mots clés: Branching random walk; spinal decomposition

Domaines: Probabilités [math.PR]

Xinxin Chen. Scaling limit of the path leading to the leftmost particle in a branching random walk. SIAM Theory of Probability and its Applications, 2013, 59 (4), pp.567-589. ⟨hal-00827040⟩