Minimal displacement of branching random walk

  • Maury D. Bramson


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.


Stochastic Process Random Walk Probability Theory Mathematical Biology Minimal Displacement 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  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

Personalised recommendations