Open Access
2010 Consistent Minimal Displacement of Branching Random Walks
Ming Fang, Ofer Zeitouni
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Electron. Commun. Probab. 15: 106-118 (2010). DOI: 10.1214/ECP.v15-1533


Let $\mathbb{T}$ denote a rooted $b$-ary tree and let $\{S_v\}_{v\in \mathbb{T}}$ denote a branching random walk indexed by the vertices of the tree, where the increments are i.i.d. and possess a logarithmic moment generating function $\Lambda(\cdot)$. Let $m_n$ denote the minimum of the variables $S_v$ over all vertices at the $n$th generation, denoted by $\mathbb{D}_n$. Under mild conditions, $m_n/n$ converges almost surely to a constant, which for convenience may be taken to be $0$. With $\bar S_v=\max\{S_w: w$ is on the geodesic connecting the root to $v \}$, define $L_n=\min_{v\in \mathbb{D}_n} \bar S_v$. We prove that $L_n/n^{1/3}$ converges almost surely to an explicit constant $l_0$. This answers a question of Hu and Shi.


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Ming Fang. Ofer Zeitouni. "Consistent Minimal Displacement of Branching Random Walks." Electron. Commun. Probab. 15 106 - 118, 2010.


Accepted: 29 March 2010; Published: 2010
First available in Project Euclid: 6 June 2016

zbMATH: 1201.60041
MathSciNet: MR2606508
Digital Object Identifier: 10.1214/ECP.v15-1533

Primary: 60G50
Secondary: 60J80

Keywords: Branching random walk , Consistent Minimal Displacement

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