Louigi Addario-Berry, Bruce Reed
Published 2007-12-16, updated 2009-07-24Version 3
Given a branching random walk, let $M_n$ be the minimum position of any member of the $n$th generation. We calculate $\mathbf{E}M_n$ to within O(1) and prove exponential tail bounds for $\mathbf{P}\{|M_n-\mathbf{E}M_n|>x\}$, 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 $\mathbf {E}M_n$ when the branching random walk has bounded branching and step size.
Comments: Published in at http://dx.doi.org/10.1214/08-AOP428 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)
Journal: Annals of Probability 2009, Vol. 37, No. 3, 1044-1079
Consistent Minimal Displacement of Branching Random Walks
Limit theorems for the minimal position of a branching random walk in random environment
Maximal displacement of a branching random walk in time-inhomogeneous environment
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