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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

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    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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