Minimal displacement of branching random walk

  • Maury D. Bramson
Article

Summary

Let\(\mathfrak{X}\) denote a branching random walk in\(\mathbb{R}^1 \) with mean particle productionm, m>1, and with incremental spatial distributionG, withG({0}) =p andG({1})=1−p. Ifmp=1, then the minimal displacement of\(\mathfrak{X}\) behaves asymptotically like log logn/log 2. If the conditionG({1})=1−p is replaced byG((0, ∞))=1−p, we obtain a similar result.

Keywords

Stochastic Process Random Walk Probability Theory Mathematical Biology Minimal Displacement 

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References

  1. 1.
    Athreya, K.B., Ney, P.E.: Branching Processes. Berlin-Heidelberg-New York: Springer 1972Google Scholar
  2. 2.
    Cohn, H.: Almost sure convergence of branching processes. Z. Wahrscheinlichkeitstheorie verw. Gebiete38, 73–81 (1977)Google Scholar
  3. 3.
    Darling, D.A.: The Galton-Watson process with infinite mean. J. App. Probability7, 455–456 (1970)Google Scholar
  4. 4.
    Hammersley, J.M.: Postulates for subadditive processes. Ann. Probability2, 652–680 (1974)Google Scholar
  5. 5.
    Harris, T.E.: The Theory of Branching Processes. Berlin-Heidelberg-New York: Springer 1963Google Scholar
  6. 6.
    Joffe, A., Le Cam, L., Neveu, J.: Sur la loi des grands nombres pour des variables aléatoires de Bernoulli attachées à un arbre dyadique. C.R. Acad. Sci., Paris, Série A277, 963–964 (1973)Google Scholar
  7. 7.
    Loève, M.: Probability Theory, 3rd ed. Princeton: Van Nostrand, 1963Google Scholar

Copyright information

© Springer-Verlag 1978

Authors and Affiliations

  • Maury D. Bramson
    • 1
  1. 1.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA

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