Journal of Applied Probability

Tightness for maxima of generalized branching random walks

Ming Fang

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Abstract

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

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

Dates
First available in Project Euclid: 6 September 2012

Permanent link to this document
https://projecteuclid.org/euclid.jap/1346955324

Digital Object Identifier
doi:10.1239/jap/1346955324

Mathematical Reviews number (MathSciNet)
MR3012090

Zentralblatt MATH identifier
1261.60097

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

Keywords
Branching random walk recursion tightness

Citation

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


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References

  • Addario-Berry, L. and Reed, B. A. (2009). Minima in branching random walks. Ann. Prob. 37, 1044–1079.
  • Bramson, M. D. (1978). Maximal displacement of branching Brownian motion. Commun. Pure Appl. Math. 31, 531–581.
  • Bramson, M. and Zeitouni, O. (2009). Tightness for a family of recursion equations. Ann. Prob. 37, 615–653.
  • Bramson, M. and Zeitouni, O. (2012). Tightness of the recentered maximum of the two-dimensional discrete Gaussian free field. Commun. Pure Appl. Math. 65, 1–20.
  • Dekking, F. M. and Host, B. (1991). Limit distributions for minimal displacement of branching random walks. Prob. Theory Relat. Fields 90, 403–426.