Minima in branching random walks

Details Minima in branching random walks Minima in branching random walks

2007-12-16

arxiv.org/abs/0712.2582v3

Annals of Probability 2009, Vol. 37, No. 3, 1044-1079

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/08-AOP428 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of

Scientific paper

10.1214/08-AOP428

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.

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