Probability Theory and Related Fields

, Volume 90, Issue 3, pp 403-426

First online:

Limit distributions for minimal displacement of branching random walks

  • F. M. DekkingAffiliated withDepartment of Mathematics and Informatics, Delft University of Technology
  • , B. HostAffiliated withDépartement de Mathématique et Informatique, Faculté des Sciences de Luminy

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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.