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Maxima of branching random walks

  • Richard Durrett
Article

Summary

In recent years several authors have obtained limit theorems for L n , the location of the rightmost particle in a supercritical branching random walk but all of these results have been proved under the assumption that the offspring distribution has ϕ(θ) = ∝ exp(θx)dF(x)<∞ for some θ>0. In this paper we investigate what happens when there is a slowly varying function K so that 1−F(x)x }-q K(x) as x → ∞ and log(−x)F(x)→0 as x→−∞. In this case we find that there is a sequence of constants a n , which grow exponentially, so that L n /a n converges weakly to a nondegenerate distribution. This result is in sharp contrast to the linear growth of L n observed in the case ϕ(θ)<∞.

Keywords

Stochastic Process Random Walk Probability Theory Limit Theorem Mathematical Biology 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 1983

Authors and Affiliations

  • Richard Durrett
    • 1
  1. 1.Dept. of MathematicsUniversity of CaliforniaLos AngelesUSA

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