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Stable-like fluctuations of Biggins’ martingales

Author

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  • Iksanov, Alexander
  • Kolesko, Konrad
  • Meiners, Matthias

Abstract

Let (Wn(θ))n∈N0 be Biggins’ martingale associated with a supercritical branching random walk, and let W(θ) be its almost sure limit. Under a natural condition for the offspring point process in the branching random walk, we show that if the law of W1(θ) belongs to the domain of normal attraction of an α-stable distribution for some α∈(1,2), then, as n→∞, there is weak convergence of the tail process (W(θ)−Wn−k(θ))k∈N0, properly normalized, to a random scale multiple of a stationary autoregressive process of order one with α-stable marginals.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:129:y:2019:i:11:p:4480-4499
    DOI: 10.1016/j.spa.2018.11.022
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    References listed on IDEAS

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    1. Biggins, J. D., 1998. "Lindley-type equations in the branching random walk," Stochastic Processes and their Applications, Elsevier, vol. 75(1), pages 105-133, June.
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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).
    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.

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