The Annals of Probability

Minima in branching random walks

Louigi Addario-Berry and Bruce Reed

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Given a branching random walk, let Mn be the minimum position of any member of the nth generation. We calculate EMn to within O(1) and prove exponential tail bounds for P{|MnEMn|>x}, under quite general conditions on the branching random walk. In particular, together with work by Bramson [Z. Wahrsch. Verw. Gebiete 45 (1978) 89–108], our results fully characterize the possible behavior of EMn when the branching random walk has bounded branching and step size.

Article information

Ann. Probab., Volume 37, Number 3 (2009), 1044-1079.

First available in Project Euclid: 19 June 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Branching random walks branching processes random trees


Addario-Berry, Louigi; Reed, Bruce. Minima in branching random walks. Ann. Probab. 37 (2009), no. 3, 1044--1079. doi:10.1214/08-AOP428.

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