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Regular variation of fixed points of the smoothing transform

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  • Liang, Xingang
  • Liu, Quansheng

Abstract

Let (N,A1,A2,…) be a sequence of random variables with N∈N∪{∞} and Ai∈R+. We are interested in asymptotic properties of non-negative solutions of the distributional equation Z=(d)∑i=1NAiZi, where Zi are non-negative random variables independent of each other and independent of (N,A1,A2,…), each having the same distribution as Z which is unknown. For a solution Z with finite mean, we prove that for a given α>1, P(Z>x) is a function regularly varying at ∞ of index −α if and only if the same is true for P(Y1>x), where Y1=∑i=1NAi. The result completes the sufficient condition obtained by Iksanov & Polotskiy (2006) on the branching random walk. A similar result on sufficient condition is also established for the case where α=1.

Suggested Citation

  • Liang, Xingang & Liu, Quansheng, 2020. "Regular variation of fixed points of the smoothing transform," Stochastic Processes and their Applications, Elsevier, vol. 130(7), pages 4104-4140.
    • Handle: RePEc:eee:spapps:v:130:y:2020:i:7:p:4104-4140
    DOI: 10.1016/j.spa.2019.11.011
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    References listed on IDEAS

    as
    1. 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.
    2. Iksanov, Alexander & Kolesko, Konrad & Meiners, Matthias, 2019. "Stable-like fluctuations of Biggins’ martingales," Stochastic Processes and their Applications, Elsevier, vol. 129(11), pages 4480-4499.
    3. 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. Huang, Chunmao & Liu, Quansheng, 2024. "Limit theorems for a branching random walk in a random or varying environment," Stochastic Processes and their Applications, Elsevier, vol. 172(C).

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