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Abstract: Given a branching random walk, let $M n$ be the minimum position of anymember of the $n$th generation. We calculate $\mathbf{E}M n$ to within O1 andprove exponential tail bounds for $\mathbf{P}\{|M n-\mathbf{E}M n|>x\}$, underquite general conditions on the branching random walk. In particular, togetherwith work by Bramson Z. Wahrsch. Verw. Gebiete 45 1978 89-108, our resultsfully characterize the possible behavior of $\mathbf {E}M n$ when the branchingrandom walk has bounded branching and step size.

Author: Louigi Addario-Berry, Bruce Reed


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