Maxima of branching random walks

  • Richard Durrett


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)<∞.


Stochastic Process Random Walk Probability Theory Limit Theorem Mathematical Biology 
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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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