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Maximal displacement of a supercritical branching random walk in a time-inhomogeneous random environment

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  • Mallein, Bastien
  • Miłoś, Piotr

Abstract

The behavior of the maximal displacement of a supercritical branching random walk has been a subject of intense studies for a long time. But only recently the case of time-inhomogeneous branching has gained focus. The contribution of this paper is to analyze a time-inhomogeneous model with two levels of randomness. In the first step a sequence of branching laws is sampled independently according to a distribution on the set of point measures’ laws. Conditionally on the realization of this sequence (called environment) we define a branching random walk and find the asymptotic behavior of its maximal particle. It is of the form Vn−φlogn+oP(logn), where Vn is a function of the environment that behaves as a random walk and φ>0 is a deterministic constant, which turns out to be bigger than the usual logarithmic correction of the homogeneous branching random walk.

Suggested Citation

  • Mallein, Bastien & Miłoś, Piotr, 2019. "Maximal displacement of a supercritical branching random walk in a time-inhomogeneous random environment," Stochastic Processes and their Applications, Elsevier, vol. 129(9), pages 3239-3260.
    • Handle: RePEc:eee:spapps:v:129:y:2019:i:9:p:3239-3260
    DOI: 10.1016/j.spa.2018.09.008
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    References listed on IDEAS

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    1. Hu, Yueyun & Yoshida, Nobuo, 2009. "Localization for branching random walks in random environment," Stochastic Processes and their Applications, Elsevier, vol. 119(5), pages 1632-1651, May.
    2. Mallein, Bastien, 2015. "Maximal displacement of a branching random walk in time-inhomogeneous environment," Stochastic Processes and their Applications, Elsevier, vol. 125(10), pages 3958-4019.
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