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Asymptotic properties and absolute continuity of laws stable by random weighted mean

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  • Liu, Quansheng

Abstract

We study properties of stable-like laws, which are solutions of the distributional equation where (N,A1,A2,...) is a given random variable with values in {0,1,...}x[0,[infinity])x[0,[infinity])x..., and Z,Z1,Z2,... are identically distributed positive random variables, independent of each other and independent of (N,A1,A2,...). Examples of such laws contain the laws of the well-known limit random variables in: (a) the Galton-Watson process or general branching processes, (b) branching random walks, (c) multiplicative processes, and (d) smoothing processes. For any solution Z (with finite or infinite mean), we find asymptotic properties of the distribution function P(Z[less-than-or-equals, slant]x) and those of the characteristic function EeitZ; we prove that the distribution of Z is absolutely continuous on (0,[infinity]), and that its support is the whole half-line [0,[infinity]). Solutions which are not necessarily positive are also considered.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:95:y:2001:i:1:p:83-107
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    References listed on IDEAS

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    1. 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.
    2. Liu, Quansheng, 2000. "On generalized multiplicative cascades," Stochastic Processes and their Applications, Elsevier, vol. 86(2), pages 263-286, April.
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    Cited by:

    1. Quansheng Liu & Emmanuel Rio & Alain Rouault, 2003. "Limit Theorems for Multiplicative Processes," Journal of Theoretical Probability, Springer, vol. 16(4), pages 971-1014, October.
    2. Bertoin, Jean, 2006. "Different aspects of a random fragmentation model," Stochastic Processes and their Applications, Elsevier, vol. 116(3), pages 345-369, March.
    3. Roozegar, Rasool & Soltani, A.R., 2015. "On the asymptotic behavior of randomly weighted averages," Statistics & Probability Letters, Elsevier, vol. 96(C), pages 269-272.
    4. Xiaoqiang Wang & Chunmao Huang, 2017. "Convergence of Martingale and Moderate Deviations for a Branching Random Walk with a Random Environment in Time," Journal of Theoretical Probability, Springer, vol. 30(3), pages 961-995, September.
    5. 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.
    6. Caliebe, Amke & Rösler, Uwe, 2003. "Fixed points with finite variance of a smoothing transformation," Stochastic Processes and their Applications, Elsevier, vol. 107(1), pages 105-129, September.
    7. Yang, Hairuo, 2023. "On the law of terminal value of additive martingales in a remarkable branching stable process," Stochastic Processes and their Applications, Elsevier, vol. 158(C), pages 361-376.
    8. Caputo, Pietro & Quattropani, Matteo, 2021. "Mixing time trichotomy in regenerating dynamic digraphs," Stochastic Processes and their Applications, Elsevier, vol. 137(C), pages 222-251.
    9. Bassetti, Federico & Matthes, Daniel, 2014. "Multi-dimensional smoothing transformations: Existence, regularity and stability of fixed points," Stochastic Processes and their Applications, Elsevier, vol. 124(1), pages 154-198.
    10. 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.
    11. Bassetti, Federico & Ladelli, Lucia, 2023. "Central limit theorem in uniform metrics for generalized Kac equations," Stochastic Processes and their Applications, Elsevier, vol. 166(C).

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