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Minima in branching random walks. (English) Zbl 1196.60142

The authors investigate a supercritical branching random walk: A Galton-Watson tree where each edge is supplemented with a (iid) weight and where each node is labeled by the sum of weights on the edges from the node to the root. Let \(M(n)\) be the minimum over the labels of all nodes in counting distance \(n\) from the root. The main result of the paper is a characterization of the asymptotic behaviour of the mean \(E(M(n))\) conditioned on that the branching random walk survives. For the case of bounded branching and weight size the results complement results from the literature such that the behaviour of the mean \(E(M(n))\) is completely characterized.

MSC:

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