Summary
We study the minimal displacement (X n ) of branching random walk with non-negative steps. It is shown that (X n −EX n ) is tight under a mild moment condition on the displacements. For supercritical B.R.W. (X n ) converges almost surely. For critical B.R.W. we determine the possible limit points of (X n −EX n ), and we prove a generalization of Kolmogorov's theorem on the extinction probability of a critical branching process. Finally we generalize Bramson's results on the almost sure convergence ofX n log 2/log logn.
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Dekking, F.M., Host, B. Limit distributions for minimal displacement of branching random walks. Probab. Th. Rel. Fields 90, 403–426 (1991). https://doi.org/10.1007/BF01193752
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DOI: https://doi.org/10.1007/BF01193752