A Simple Path to Biggins’ Martingale Convergence for Branching Random Walk

  • Russell Lyons
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 84)

Abstract

We give a simple non-analytic proof of Biggins’ theorem on martingale convergence for branching random walks.

Key words

Galton-Watson 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Biggins, J. D. (1977) Martingale convergence in the branching random walk. J. Appl. Prob. 1425–37.MathSciNetMATHCrossRefGoogle Scholar
  2. Chauvin, B. and Rouault, A. (1988) KPP equation and supercritical branching Brownian motion in the subcritical speed area. Application to spatial trees, Probab. Theory Relat. Fields 80299–314.MathSciNetMATHCrossRefGoogle Scholar
  3. Chauvin, B., Rouault, A., and Wakolbinger, A. (1991) Growing conditioned trees, Stochastic Process. Appl. 39 117–130.MathSciNetMATHCrossRefGoogle Scholar
  4. Durrett, R. (1991) Probability: Theory and Examples. Wadsworth, Pacific Grove, California.Google Scholar
  5. Hawkes, J. (1981) Trees generated by a simple branching process, J. London Math. Soc. 24 373–384.MathSciNetMATHCrossRefGoogle Scholar
  6. Joffe, A. and Waugh, W. A. O’N. (1982) Exact distributions of kin numbers in a Galton-Watson process, J. Appl. Prob. 19767–775.MathSciNetMATHCrossRefGoogle Scholar
  7. Kahane, J.-P. and Peyrière, J. (1976) Sur certaines martingales de Benoit Mandelbrot, Adv. in Math. 22131–145.MATHCrossRefGoogle Scholar
  8. Kauenberg, O. (1977) Stability of critical cluster fields, Math. Nachr. 777–43.MathSciNetCrossRefGoogle Scholar
  9. Kesten, H. (1986) Subdiffusive behavior of random walk on a random cluster, Ann. Inst. H. PoincaréProbab. Statist. 22425–487.Google Scholar
  10. Lyons, R., Pemantle, R. Peres, Y. (1995) Conceptual proofs of L log L criteria for mean behavior of branching processes, Ann. Probab. 23125–1138.MathSciNetMATHCrossRefGoogle Scholar
  11. Rouault, A. (1981) Lois empiriques dans les processus de branchement spatiaux ho-mogènes supercritiques, C. R. Acad. Sci. Paris S¨¦r. I. Math. 292933–936.MathSciNetMATHGoogle Scholar
  12. Waymire, E. C. and Williams, S. C. (1996) A cascade decomposition theory with ap-plications to Markov and exchangeable cascades, Trans. Amer. Math. Soc. 348585–632.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1997

Authors and Affiliations

  • Russell Lyons
    • 1
  1. 1.Department of MathematicsIndiana UniversityBloomington

Personalised recommendations