Electronic Communications in Probability

Consistent Minimal Displacement of Branching Random Walks

Ming Fang and Ofer Zeitouni

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Branching Random Walk Consistent Minimal Displacement

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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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  • Addario-Berry, Louigi; Reed, Bruce. Minima in branching random walks. Ann. Probab. 37 (2009), no. 3, 1044–1079.
  • Bramson, Maury D. Maximal displacement of branching Brownian motion. Comm. Pure Appl. Math. 31 (1978), no. 5, 531–581.
  • Bramson, Maury D. Minimal displacement of branching random walk. Z. Wahrsch. Verw. Gebiete 45 (1978), no. 2, 89–108.
  • Dembo, Amir; Zeitouni, Ofer. Large deviations techniques and applications.Second edition.Applications of Mathematics (New York), 38. Springer-Verlag, New York, 1998. xvi+396 pp. ISBN: 0-387-98406-2
  • Hu, Yueyun; Shi, Zhan. Slow movement of random walk in random environment on a regular tree. Ann. Probab. 35 (2007), no. 5, 1978–1997.
  • G. Faraud, Y. Hu and Z. Shi, An almost sure convergence for stochastically biased random walks on trees, personal communication (2009).
  • MogulʹskiÄ­, A. A. Small deviations in the space of trajectories.(Russian) Teor. Verojatnost. i Primenen. 19 (1974), 755–765.