# Minima in branching random walks

1 MASCOTTE - Algorithms, simulation, combinatorics and optimization for telecommunications
CRISAM - Inria Sophia Antipolis - Méditerranée , Laboratoire I3S - COMRED - COMmunications, Réseaux, systèmes Embarqués et Distribués
Abstract : Given a branching random walk, let $M_n$ be the minimum position of any member of the $n$th generation. We calculate $\\mathbfEM_n$ to within O(1) and prove exponential tail bounds for $\\mathbfP{|M_n-\\mathbfEM_n|>x}$, 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 $\\mathbf EM_n$ when the branching random walk has bounded branching and step size.
Document type :
Journal articles
Domain :

https://hal.inria.fr/hal-00795281
Contributor : Alain Monteil Connect in order to contact the contributor
Submitted on : Wednesday, February 27, 2013 - 5:16:25 PM
Last modification on : Monday, October 12, 2020 - 10:30:26 AM

### Identifiers

• HAL Id : hal-00795281, version 1

### Citation

Louigi Addario-Berry, Bruce Reed. Minima in branching random walks. Annals of Probability, Institute of Mathematical Statistics, 2009, 37, pp.1044―1079. ⟨hal-00795281⟩

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