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 Electronic Journal of Probability > Vol. 14 (2009) > Paper 10 open journal systems 


A Log-Type Moment Result for Perpetuities and Its Application to Martingales in Supercritical Branching Random Walks

Gerold Alsmeyer, Institut fur Mathematische Statistik, Munster University
Alex Iksanov, Faculty of Cybernetics, National T.Shevchenko University of Kiev


Abstract
Infinite sums of i.i.d. random variables discounted by a multiplicative random walk are called perpetuities and have been studied by many authors. The present paper provides a log-type moment result for such random variables under minimal conditions which is then utilized for the study of related moments of a.s. limits of certain martingales associated with the supercritical branching random walk. The connection arises upon consideration of a size-biased version of the branching random walk originally introduced by Lyons. As a by-product, necessary and sufficient conditions for uniform integrability of these martingales are provided in the most general situation which particularly means that the classical (LlogL)-condition is not always needed.


Full text: PDF

Pages: 289-313

Published on: January 29, 2009
Bibliography
  1. G. Alsmeyer. On generalized renewal measures and certain first passage times. Ann. Probab 20 (1992), 1229-1247. MR1175261 (93i:60089)
  2. G. Alsmeyer, A. Iksanov and U. R"{o}sler. On distributional properties of perpetuities. J. Theoret. Probab. 22 (2009), to appear. Math. Review number not available.
  3. G. Alsmeyer, D. Kuhlbusch. Double martingale structure and existence of $phi$-moments for weighted branching processes. Preprint no. 08/05-S. (2006). Universit"{a}t M"{u}nster, Germany. Math. Review number not available.
  4. G. Alsmeyer, U. R"{o}sler. On the existence of $phi$-moments of the limit of a normalized supercritical Galton-Watson process. J.Theoret. Probab. 17 (2004), 905-928. MR2105740 (2005h:60252)
  5. J.D. Biggins. Martingale convergence in the branching random walk. J.Appl.Probab. 14 (1977), 25-37. MR0433619 (55 #6592)
  6. J.D. Biggins. Growth rates in the branching random walk. Z. Wahrsch. verw. Gebiete 48 (1979), 17-34. MR0533003 (80e:60095)
  7. J.D. Biggins, A.E. Kyprianou. Seneta-Heyde norming in the branching random walk. Ann.Probab. 25 (1997), 337-360. MR1428512 (98a:60118)
  8. J.D. Biggins, A.E. Kyprianou. Measure change in multitype branching. Adv. Appl. Probab. 36 (2004), 544-581. MR2058149 (2005f:60179)
  9. N.H. Bingham, R.A. Doney. Asymptotic properties of supercritical branching processes II: Crump-Mode and Jirina processes. Adv. Appl. Probab. 7 (1975), 66-82. MR0378125 (51 #14294)
  10. N.H. Bingham, C.M. Goldie and J.L. Teugels. Regular variation. Encyclopedia of Mathematics and its Applications 27 (1989) Cambridge University Press, Cambridge. MR1015093 (90i:26003)
  11. Y.S. Chow, H. Teicher. Probability theory: independence, interchangeability, martingales. Springer Texts in Statistics (1988), Springer-Verlag, New York. MR0953964 (89e:60001)
  12. K.B. Erickson. The strong law of large numbers when the mean is undefined. Trans. Amer. Math. Soc. 185 (1973), 371-381. MR0336806(49 #1579)
  13. C.M. Goldie, R.A. Maller. Stability of perpetuities. Ann. Probab. 28 (2000), 1195-1218. MR1797309 (2003b:60045)
  14. S. Harris, M.Roberts. Measure changes with extinction. Preprint no. 0811. 1696, available at www.arXiv.org, (2008). Math. Review number not available.
  15. A.M. Iksanov. Elementary fixed points of the BRW smoothing transforms with infinite number of summands. Stoch. Proc. Appl. 114 (2004), 27-50. MR2094146 (2005i:60168)
  16. A.M. Iksanov. On the rate of convergence of a regular martingale related to the branching random walk. Ukrainian Math. J. 58 (2006), 368-387. MR2271973 (2007m:60115)
  17. A.M. Iksanov, P. Negadajlov. On the supremum of a martingale associated with a branching random walk. Th. Probab. Math. Statist. 74 (2007), 49-57. MR2336778 (2008g:60257)
  18. A.M. Iksanov, U. R"{o}sler. Some moment results about the limit of a martingale related to the supercritical branching random walk and perpetuities. Ukrainian Math. J.. 58 (2006), 505-528. MR2272796 (2007j:60142)
  19. S. Janson. Moments for first-passage and last-exit times, the minimum, and related quantities for random walks with positive drift. Adv. Appl. Probab. 18 (1986), 865-879. MR0867090(88b:60164)
  20. H. Kesten, R.A. Maller. Two renewal theorems for general random walks tending to infinity. Probab. Theory Relat. Fields. 106 (1996), 1-38. MR1408415(97i:60113)
  21. H. Kesten, R.A. Stigum. A limit theorem for multidimensional Galton-Watson processes. Ann. Math. Stat. 37 (1966), 1211-1223. MR0198552(33 #6707)
  22. J.F.C. Kingman. The first birth problem for an age-dependent branching process. Ann. Probab. 3 (1975), 790-801. MR0400438 (53 #4271)
  23. D. Kuhlbusch. On weighted branching processes in random environment. Stoch. Proc. Appl. 109 (2004), 113-144. MR2024846(2004j:60088)
  24. Q. Liu. Sur une '{e}quation fonctionnelle et ses applications: une extension du th'{e}or`{e}me de Kesten-Stigum concernant des processus de branchement. Adv. Appl. Probab. 29 (1997), 353-373. MR1450934 (98j:60124)
  25. Q. Liu. On generalized multiplicative cascades. Stoch. Proc. Appl. 86 (2000), 263-286. MR1741808 (2001b:60102)
  26. R. Lyons. A simple path to Biggins' martingale convergence for branching random walk. In Athreya, K.B., Jagers, P. (eds.). Classical and Modern Branching Processes, IMA Volumes in Mathematics and its Applications 84 Springer, Berlin, 217-221. MR1601749
  27. R. Lyons., R. Pemantle and Y. Peres. Conceptual proofs of $Llog L$ criteria for mean behavior of branching processes. Ann. Probab. 23 (1995), 1125-1138. MR1349164 (96m:60194)
  28. U. R"{o}sler, V.A.Topchii and V.A. Vatutin. Convergence conditions for the weighted branching process. Discrete Math. Appl. 10 (2000), 5-21. MR1778763 (2001k:60123)
















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Electronic Journal of Probability. ISSN: 1083-6489