Journal of Applied Probability

Tightness for maxima of generalized branching random walks

Ming Fang

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We study generalized branching random walks on the real line R that allow time dependence and local dependence between siblings. Specifically, starting from one particle at time 0, the system evolves such that each particle lives for one unit amount of time, gives birth independently to a random number of offspring according to some branching law, and dies. The offspring from a single particle are assumed to move to new locations on R according to some joint displacement distribution; the branching laws and displacement distributions depend on time. At time n, Fn(·) is used to denote the distribution function of the position of the rightmost particle in generation n. Under appropriate tail assumptions on the branching laws and offspring displacement distributions, we prove that Fn(· - Med(Fn)) is tight in n, where Med(Fn) is the median of Fn. The main part of the argument is to demonstrate the exponential decay of the right tail 1 - Fn(· - Med(Fn)).

Article information

J. Appl. Probab., Volume 49, Number 3 (2012), 652-670.

First available in Project Euclid: 6 September 2012

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Digital Object Identifier

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 recursion tightness


Fang, Ming. Tightness for maxima of generalized branching random walks. J. Appl. Probab. 49 (2012), no. 3, 652--670. doi:10.1239/jap/1346955324.

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