On the empty balls of a critical or subcritical branching random walk

IF 1.2 4区 数学 Q1 MATHEMATICS Acta Mathematica Scientia Pub Date : 2024-08-27 DOI:10.1007/s10473-024-0525-0
Shuxiong Zhang, Jie Xiong
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Abstract

Let {Zn}n≥0 be a critical or subcritical d-dimensional branching random walk started from a Poisson random measure whose intensity measure is the Lebesugue measure on ℝd. Denote by Rn:= sup{u > 0: Zn({x ∈ ℝd: ∣x∣ < u}) = 0} the radius of the largest empty ball centered at the origin of Zn. In this work, we prove that after suitable renormalization, Rn converges in law to some non-degenerate distribution as n → ∈. Furthermore, our work shows that the renormalization scales depend on the offspring law and the dimension of the branching random walk. This completes the results of Révész [13] for the critical binary branching Wiener process.

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关于临界或亚临界分支随机游走的空球
设 {Zn}n≥0 是一个临界或亚临界 d 维分支随机游走,从一个强度度量为ℝd 上的勒布苏格度量的泊松随机度量开始。用 Rn:= sup{u > 0: Zn({x ∈ ℝd: ∣x∣ < u}) = 0} 表示以 Zn 的原点为中心的最大空球的半径。在这项工作中,我们证明了经过适当的重正化后,Rn 在 n →∈ 时收敛于某种非退化分布的规律。此外,我们的研究还表明,重正化尺度取决于子代规律和分支随机游走的维度。这完善了 Révész [13] 对临界二元分支维纳过程的研究结果。
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来源期刊
CiteScore
2.00
自引率
10.00%
发文量
2614
审稿时长
6 months
期刊介绍: Acta Mathematica Scientia was founded by Prof. Li Guoping (Lee Kwok Ping) in April 1981. The aim of Acta Mathematica Scientia is to present to the specialized readers important new achievements in the areas of mathematical sciences. The journal considers for publication of original research papers in all areas related to the frontier branches of mathematics with other science and technology.
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