# zbMATH — the first resource for mathematics

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.

##### MSC:
 60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
Full Text: