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Minimal positions in a branching random walk. (English) Zbl 0836.60089
Particles in a supercritical Galton-Watson process have positions attached to them in the following way. The initial ancestor is at the origin; the first generation positions are given by a point process \(Z\); each \(n\)th generation person’s family are placed, relative to the parent, using an independent copy of \(Z\). Denote by \(B_n\) the position of the leftmost particle in the \(n\)th generation. If \(\varphi (\theta)\) is the Laplace transform of the intensity measure of \(Z\), then, provided this is finite for some positive \(\theta\), \(B_n/n\) converges to a constant \(\gamma\). For branching Brownian motion M. Bramson [Commun. Pure Appl. Math. 31, 531-581 (1978; Zbl 0361.60052)] obtained much more precise results. Here a step towards an analogue of these more precise results of Bramson is taken. It is shown, roughly, that the distribution of \(B_n - n \gamma - c \log n\) has, for a suitable \(c\), an exponentially decaying left tail, and with a different \(c\), an exponentially decaying right tail.

60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
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