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
{"title":"On the empty balls of a critical or subcritical branching random walk","authors":"Shuxiong Zhang, Jie Xiong","doi":"10.1007/s10473-024-0525-0","DOIUrl":null,"url":null,"abstract":"<p>Let {<i>Z</i><sub><i>n</i></sub>}<sub><i>n</i>≥0</sub> be a critical or subcritical <i>d</i>-dimensional branching random walk started from a Poisson random measure whose intensity measure is the Lebesugue measure on ℝ<sup><i>d</i></sup>. Denote by <i>R</i><sub><i>n</i></sub>:= sup{<i>u</i> &gt; 0: <i>Z</i><sub><i>n</i></sub>({<i>x</i> ∈ ℝ<sup><i>d</i></sup>: ∣<i>x</i>∣ &lt; <i>u</i>}) = 0} the radius of the largest empty ball centered at the origin of <i>Z</i><sub><i>n</i></sub>. In this work, we prove that after suitable renormalization, <i>R</i><sub><i>n</i></sub> converges in law to some non-degenerate distribution as <i>n</i> → ∈. 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.</p>","PeriodicalId":50998,"journal":{"name":"Acta Mathematica Scientia","volume":"369 1","pages":""},"PeriodicalIF":1.2000,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Acta Mathematica Scientia","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10473-024-0525-0","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0

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.

查看原文
分享 分享
微信好友 朋友圈 QQ好友 复制链接
本刊更多论文
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。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 去求助
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.
期刊最新文献
Lévy area analysis and parameter estimation for fOU processes via non-geometric rough path theory Heat kernel on Ricci shrinkers (II) Variational analysis for the maximal time function in normed spaces Toeplitz operators between weighted Bergman spaces over the half-plane Global unique solutions for the incompressible MHD equations with variable density and electrical conductivity
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
现在去查看 取消
×
提示
确定
0
微信
客服QQ
Book学术公众号 扫码关注我们
反馈
×
意见反馈
请填写您的意见或建议
请填写您的手机或邮箱
已复制链接
已复制链接
快去分享给好友吧!
我知道了
×
扫码分享
扫码分享
Book学术官方微信
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术
文献互助 智能选刊 最新文献 互助须知 联系我们:info@booksci.cn
Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。
Copyright © 2023 Book学术 All rights reserved.
ghs 京公网安备 11010802042870号 京ICP备2023020795号-1