Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-23T15:40:20.989Z Has data issue: false hasContentIssue false
  • English
  • Français

Exponential rate of almost-sure convergence of intrinsic martingales in supercritical branching random walks

Part of: Special processes Stochastic processes Markov processes

Published online by Cambridge University Press:  14 July 2016

Alexander Iksanov  and
Matthias Meiners
Alexander Iksanov*
Affiliation:
National Taras Shevchenko University of Kyiv
Matthias Meiners*
Affiliation:
Universität Münster
*
∗Postal address: Faculty of Cybernetics, National Taras Shevchenko University of Kyiv, 01033 Kiev, Ukraine. Email address: [email protected]
∗∗Current address: Department of Mathematics, Uppsala University, 75106 Uppsala, Sweden.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We provide sufficient conditions which ensure that the intrinsic martingale in the supercritical branching random walk converges exponentially fast to its limit. We include in particular the case of Galton-Watson processes so that our results can be seen as a generalization of a result given in the classical treatise by Asmussen and Hering (1983). As an auxiliary tool, we prove ultimate versions of two results concerning the exponential renewal measures which may be of interest in themselves and which correct, generalize, and simplify some earlier works.

Keywords

Branching random walkmartingalerate of convergencerenewal theory

MSC classification

Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60K05: Renewal theory 60G42: Martingales with discrete parameter
Type
Research Article
Information
Journal of Applied Probability , Volume 47 , Issue 2 , June 2010 , pp. 513 - 525
Copyright
Copyright © Applied Probability Trust 2010 

Footnotes

This work was commenced while A. Iksanov was visiting Münster in January 2009. A Iksanov thanks G. Alsmeyer, M. Meiners, and Institut für Mathematische Statistik for Invitation, hospitality, and financial support.

References

[1] Alsmeyer, G. and Iksanov, A. (2009). A log-type moment result for perpetuities and its application to martingales in supercritical branching random walks. Electron. J. Prob. 14, 289–312.CrossRefGoogle Scholar
[2] Alsmeyer, G., Iksanov, A., Rösler, U. and Polotsky, S. (2009). Exponential rate of { L}p-convergence of intrinsic martingales in supercritical branching random walks. Theory Stoch. Process. 15, 1–18.Google Scholar
[3] Asmussen, S. and Hering, H. (1983). Branching Processes (Progress Prob. Statist. 3). Birkhäuser, Boston, MA.Google Scholar
[4] Beljaev, Ju. K. and Maksimov, V. M. (1963). Analytical properties of a generating function for the number of renewals. Theory Prob. Appl. 8, 108–112.Google Scholar
[5] Biggins, J. D. (1977). Martingale convergence in the branching random walk. J. Appl. Prob. 14, 25–37.CrossRefGoogle Scholar
[6] Biggins, J. D. and Kyprianou, A. E. (1997). Seneta-Heyde norming in the branching random walk. Ann. Prob. 25, 337–360.Google Scholar
[7] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1989). Regular Variation (Encyclopaedia Math. Appl. 27). Cambridge University Press.Google Scholar
[8] Borovkov, A. A. (1962). New limit theorems in boundary-value problems for sums of independent terms. Sibirsk. Mat. Ž. 3, 645–694.Google Scholar
[9] Gatzouras, D. (2000). On the lattice case of an almost-sure renewal theorem for branching random walks. Adv. Appl. Prob. 32, 720–737.CrossRefGoogle Scholar
[10] Heyde, C. C. (1964). Two probability theorems and their application to some first passage problems. J. Austral. Math. Soc. 4, 214–222.Google Scholar
[11] Heyde, C. C. (1966). Some renewal theorems with application to a first passage problem. Ann. Math. Statist. 37, 699–710.Google Scholar
[12] Iksanov, A. M. (2006). On the rate of convergence of a regular martingale related to a branching random walk. Ukrainian Math. J. 58, 368–387.CrossRefGoogle Scholar
[13] Liu, Q. (1997). Sur une équation fonctionnelle et ses applications: une extension du théorème de Kesten-Stigum concernant des processus de branchement. Adv. Appl. Prob. 29, 353–373.Google Scholar
[14] Lyons, R. (1997). A simple path to Biggins' martingale convergence for branching random walk. In Classical and Modern Branching Processes (Minneapolis, MN, 1994; IMA Vol. Math. Appl. 84), Springer, New York, pp. 217–221.Google Scholar
[15] Nerman, O. (1981). On the convergence of supercritical general ({C}-{M}-{J}) branching processes. Z. Wahrscheinlichkeitsth 57, 365–395.CrossRefGoogle Scholar