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On tails of fixed points of the smoothing transform in the boundary case

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  • Buraczewski, Dariusz

Abstract

Let {Ai} be a sequence of random positive numbers, such that only N first of them are strictly positive, where N is a finite a.s. random number. In this paper we investigate nonnegative solutions of the distributional equation , where Z,Z1,Z2,... are independent and identically distributed random variables, independent of N,A1,A2,.... We assume and (the boundary case), then it is known that all nonzero solutions have infinite mean. We obtain new results concerning behavior of their tails.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:119:y:2009:i:11:p:3955-3961
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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. Buraczewski, D. & Damek, E. & Zienkiewicz, J., 2018. "Pointwise estimates for first passage times of perpetuity sequences," Stochastic Processes and their Applications, Elsevier, vol. 128(9), pages 2923-2951.
    2. 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.
    3. 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.

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