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Asymptotic properties of supercritical age-dependent branching processes and homogeneous branching random walks

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  • Liu, Quansheng

Abstract

Let (Z(t): t[greater-or-equal, slanted]0) be a supercritical age-dependent branching process and let {Yn} be the natural martingale arising in a homogeneous branching random walk. Let Z be the almost sure limit of Z(t)/EZ(t)(t-->[infinity]) or that of Yn (n-->[infinity]). We study the following problems: (a) the absolute continuity of the distribution of Z and the regularity of the density function; (b) the decay rate (polynomial or exponential) of the left tail probability P(Z[less-than-or-equals, slant]x) as x-->0, and that of the characteristic function EeitZ and its derivative as t-->[infinity]; (c) the moments and decay rate (polynomial or exponential) of the right tail probability P(Z>x) as x-->[infinity], the analyticity of the characteristic function [phi](t)=EeitZ and its growth rate as an entire characteristic function. The results are established for non-trivial solutions of an associated functional equation, and are therefore also applicable for other limit variables arising in age-dependent branching processes and in homogeneous branching random walks.

Suggested Citation

  • Liu, Quansheng, 1999. "Asymptotic properties of supercritical age-dependent branching processes and homogeneous branching random walks," Stochastic Processes and their Applications, Elsevier, vol. 82(1), pages 61-87, July.
    • Handle: RePEc:eee:spapps:v:82:y:1999:i:1:p:61-87
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    Citations

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    Cited by:

    1. Li, Yingqiu & Liu, Quansheng & Peng, Xuelian, 2019. "Harmonic moments, large and moderate deviation principles for Mandelbrot’s cascade in a random environment," Statistics & Probability Letters, Elsevier, vol. 147(C), pages 57-65.
    2. Buraczewski, Dariusz, 2009. "On tails of fixed points of the smoothing transform in the boundary case," Stochastic Processes and their Applications, Elsevier, vol. 119(11), pages 3955-3961, November.
    3. Brigitte Chauvin & Cécile Mailler & Nicolas Pouyanne, 2015. "Smoothing Equations for Large Pólya Urns," Journal of Theoretical Probability, Springer, vol. 28(3), pages 923-957, September.
    4. Kuhlbusch, Dirk, 2004. "On weighted branching processes in random environment," Stochastic Processes and their Applications, Elsevier, vol. 109(1), pages 113-144, January.
    5. Liu, Quansheng, 2001. "Asymptotic properties and absolute continuity of laws stable by random weighted mean," Stochastic Processes and their Applications, Elsevier, vol. 95(1), pages 83-107, September.
    6. Huang, Chunmao & Liu, Quansheng, 2012. "Moments, moderate and large deviations for a branching process in a random environment," Stochastic Processes and their Applications, Elsevier, vol. 122(2), pages 522-545.

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