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<p>In this article, we consider a Branching Random Walk (BRW) on the real line. The genealogical structure is assumed to be given through a supercritical branching process in the i.i.d. environment and satisfies the Kesten-Stigum condition. The displacements coming from the same parent are assumed to have jointly regularly varying tails. Conditioned on the survival of the underlying genealogical tree, we prove that the appropriately normalized (normalization depends on the quenched size of the <i>n</i>-th generation) maximum among positions at the <i>n</i>-th generation converges weakly to a scale-mixture of Frechét random variable. Furthermore, we derive the weak limit of the point processes composed of appropriately scaled positions at the <i>n</i>-th generation and show that the limit point process is a randomly scaled scale-decorated Poisson point process. We obtain an explicit description of the limit point process. This description captures the influence of the environment in the joint asymptotic behavior of the extreme positions. We show that the law of the clusters in the limit depends on the time-reversed environment (environment experienced by the rightmost/leftmost position). The asymptotic distribution of the normalized rightmost position is derived as a consequence. This approach (based on weak convergence of extremal processes) cannot be adapted when the genealogical structure is given through a supercritical Branching Process in a Time-inhomogeneous Environment (BPTiE) (due to lack of structural regularity in the genealogical structure). We provide a simple example where the point processes do not converge weakly. The tightness of the point processes holds in this example though they do not (weakly) converge (due to having different subsequential weak limits).</p>

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Extreme positions of regularly varying branching random walk in a random and time-inhomogeneous environment

  • 期刊论文

  • 发表日期：

    2025年4月7日

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  • Ayan Bhattacharya,
  • Zbigniew Palmowski

摘要

In this article, we consider a Branching Random Walk (BRW) on the real line. The genealogical structure is assumed to be given through a supercritical branching process in the i.i.d. environment and satisfies the Kesten-Stigum condition. The displacements coming from the same parent are assumed to have jointly regularly varying tails. Conditioned on the survival of the underlying genealogical tree, we prove that the appropriately normalized (normalization depends on the quenched size of the n-th generation) maximum among positions at the n-th generation converges weakly to a scale-mixture of Frechét random variable. Furthermore, we derive the weak limit of the point processes composed of appropriately scaled positions at the n-th generation and show that the limit point process is a randomly scaled scale-decorated Poisson point process. We obtain an explicit description of the limit point process. This description captures the influence of the environment in the joint asymptotic behavior of the extreme positions. We show that the law of the clusters in the limit depends on the time-reversed environment (environment experienced by the rightmost/leftmost position). The asymptotic distribution of the normalized rightmost position is derived as a consequence. This approach (based on weak convergence of extremal processes) cannot be adapted when the genealogical structure is given through a supercritical Branching Process in a Time-inhomogeneous Environment (BPTiE) (due to lack of structural regularity in the genealogical structure). We provide a simple example where the point processes do not converge weakly. The tightness of the point processes holds in this example though they do not (weakly) converge (due to having different subsequential weak limits).