Consistent Minimal Displacement of Branching Random Walks
Ming Fang, University of Minnesota
Ofer Zeitouni, University of Minnesota and Weizmann Institute
Abstract
Let T denote a rooted b-ary tree and let {Sv}v∈ T denote a branching random walk indexed by the vertices of the tree, where the increments are i.i.d. and possess a logarithmic moment generating function Λ(.) Let mn denote the minimum of the variables Sv over all vertices at the nth generation, denoted by Dn. Under mild conditions, mn/n converges almost surely to a constant, which for convenience may be taken to be 0. With $bar Sv=max{S_w: w is on the geodesic connecting the root
to v}, define Ln=minv∈ Dn bar Sv. We prove that Ln/n1/3 converges almost surely to an explicit constant l0. This answers a question of Hu and Shi.
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