Electronic Communications in Probability

Consistent Minimal Displacement of Branching Random Walks

Ming Fang and Ofer Zeitouni

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Abstract

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.

Article information

Source
Electron. Commun. Probab., Volume 15 (2010), paper no. 11, 106-118.

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

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465243954

Digital Object Identifier
doi:10.1214/ECP.v15-1533

Mathematical Reviews number (MathSciNet)
MR2606508

Zentralblatt MATH identifier
1201.60041

Subjects
Primary: 60G50: Sums of independent random variables; random walks
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Keywords
Branching Random Walk Consistent Minimal Displacement

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Fang, Ming; Zeitouni, Ofer. Consistent Minimal Displacement of Branching Random Walks. Electron. Commun. Probab. 15 (2010), paper no. 11, 106--118. doi:10.1214/ECP.v15-1533. https://projecteuclid.org/euclid.ecp/1465243954


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References

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