Some moment results about the limit of a martingale related to the supercritical branching random walk and perpetuities

  • A. M. Iksanov
  • U. Rösler


Let \(\mathcal{M}^{(n)} \), n = 1, 2, ..., be the supercritical branching random walk in which the family sizes may be infinite with positive probability. Assume that a natural martingale related to \(\mathcal{M}^{(n)} \) converges almost surely and in the mean to a random variable W. For a large subclass of nonnegative and concave functions ƒ, we provide a criterion for the finiteness of \(\mathbb{E}\) Wf(W). The main assertions of the present paper generalize some results obtained recently in Kuhlbusch’s Ph.D. thesis as well as previously known results for the Galton-Watson processes. In the process of the proof, we study the existence of the ƒ-moments of perpetuities.


Random Walk Concave Function Borel Function Independent Copy Moment Result 
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  1. 1.
    U. Rösler, “A fixed point theorem for distributions,” Stochast. Process. Appl., 42, 195–214 (1992).zbMATHCrossRefGoogle Scholar
  2. 2.
    Q. Liu, “Fixed points of a generalized smoothing transformation and applications to the branching random walk,” Adv. Appl. Probab., 30, 85–112 (1998).zbMATHCrossRefGoogle Scholar
  3. 3.
    A. M. Iksanov, “Elementary fixed points of the BRW smoothing transforms with infinite number of summands,” Stochast. Process. Appl., 114, 27–50 (2004).zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    J. F. C. Kingman, “The first birth problem for an age-dependent branching process,” Ann. Probab., 3, 790–801 (1975).zbMATHMathSciNetGoogle Scholar
  5. 5.
    J. D. Biggins, “Martingale convergence in the branching random walk,” J. Appl. Probab., 14, 25–37 (1977).zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Q. Liu, “Sur une équation fonctionnelle et ses applications: une extension du thé orème de Kesten-Stigum concernant des processus de branchement,” Adv. Appl. Probab., 29, 353–373 (1997).zbMATHCrossRefGoogle Scholar
  7. 7.
    R. Lyons, “A simple path to Biggins’ martingale convergence for branching random walk,” in: K. B. Athreya and P. Jagers (editors), Classical and Modern Branching Processes, Springer, Berlin (1997), pp. 217–221.Google Scholar
  8. 8.
    C. M. Goldie and R. A. Maller, “Stability of perpetuities,” Ann. Probab., 28, 1195–1218 (2000).zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    M. S. Sgibnev, “On the existence of submultiplicative moments for the stationary distributions of some Markovian random walks,” J. Appl. Probab., 36, 78–85 (1999).zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    G. Alsmeyer and U. Rosler, “On the existence of θ-moments of the limit of a normalized supercritical Galton-Watson process,” J. Theor. Probab., 17, 905–928 (2004).zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    A. M. Iksanov, “On some moments of the limit random variable for a normalized supercritical Galton-Watson process,” in: L. R. Velle (editor), Focus on Probability Theory, Nova Science Publishers, New York (2004), pp. 151–158.Google Scholar
  12. 12.
    J. D. Biggins, “Growth rates in the branching random walk,” Z. Wahrscheinlichkeitstheor. Verw. Geb., 48, 17–34 (1979).zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    U. Rösler, V. A. Topchii, and V. A. Vatutin, “Convergence conditions for the weighted branching process,” Discrete Math. Appl., 10, 5–21 (2000).zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    D. Kuhlbusch, Moment Conditions for Weighted Branching Processes, PhD Thesis, Munster University, Munster (2004).zbMATHGoogle Scholar
  15. 15.
    R. Hardy and S. C. Harris, Spine Proofs for L p-Convergence of Branching-Diffusion Martingales, Mathematics Preprint 0405, University of Bath, Bath (2004); available online at∼ massch/Research/Papers/spine-Lp-cgce.pdf.Google Scholar
  16. 16.
    H. G. Kellerer, Ergodic Behaviour of Affine Recursions III: Positive Recurrence and Null Recurrence, Technical Reports, Mathematisches Institut Universität München, München (1992).Google Scholar
  17. 17.
    W. Vervaat, “On a stochastic difference equation and a representation of non-negative infinitely divisible random variables,” Adv. Appl. Probab., 11, 750–783 (1979).zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    C. M. Goldie and R. Grubel, “Perpetuities with thin tails,” Adv. Appl. Probab., 28, 463–480 (1996).zbMATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    V. F. Araman and P. W. Glynn, Tail Asymptotics for the Maximum of Perturbed Random Walk, Preprint, New York University, New York (2004); available online at∼varaman.Google Scholar
  20. 20.
    W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 2, Wiley, New York (1966).zbMATHGoogle Scholar
  21. 21.
    A. K. Grinceviícius, “A random difference equation,” Lith. Math. J., 21, 302–306 (1981).CrossRefGoogle Scholar
  22. 22.
    R. Keener, “A note on the variance of a stopping time,” Ann. Statist., 15, 1709–1712 (1987).zbMATHMathSciNetGoogle Scholar
  23. 23.
    M. S. Sgibnev, “Submultiplicative moments of the supremum of a random walk with negative drift, ” Statist. Probab. Lett., 32, 377–383 (1997).zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • A. M. Iksanov
    • 1
  • U. Rösler
    • 2
  1. 1.Shevchenko Kiev National UniversityKiev
  2. 2.University of KielKielGermany

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