Boundedness of the nodal domains of additive Gaussian fields

IF 0.4 Q4 STATISTICS & PROBABILITY
S. Muirhead
{"title":"Boundedness of the nodal domains of additive Gaussian fields","authors":"S. Muirhead","doi":"10.1090/tpms/1169","DOIUrl":null,"url":null,"abstract":"We study the connectivity of the excursion sets of additive Gaussian fields, i.e. stationary centred Gaussian fields whose covariance function decomposes into a sum of terms that depend separately on the coordinates. Our main result is that, under mild smoothness and correlation decay assumptions, the excursion sets \n\n \n \n {\n f\n ≤\n ℓ\n }\n \n \\{f \\le \\ell \\}\n \n\n of additive planar Gaussian fields are bounded almost surely at the critical level \n\n \n \n \n ℓ\n c\n \n =\n 0\n \n \\ell _c = 0\n \n\n. Since we do not assume positive correlations, this provides the first examples of continuous non-positively-correlated stationary planar Gaussian fields for which the boundedness of the nodal domains has been confirmed. By contrast, in dimension \n\n \n \n d\n ≥\n 3\n \n d \\ge 3\n \n\n the excursion sets have unbounded components at all levels.","PeriodicalId":42776,"journal":{"name":"Theory of Probability and Mathematical Statistics","volume":null,"pages":null},"PeriodicalIF":0.4000,"publicationDate":"2021-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Theory of Probability and Mathematical Statistics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1090/tpms/1169","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
引用次数: 0

Abstract

We study the connectivity of the excursion sets of additive Gaussian fields, i.e. stationary centred Gaussian fields whose covariance function decomposes into a sum of terms that depend separately on the coordinates. Our main result is that, under mild smoothness and correlation decay assumptions, the excursion sets { f ≤ ℓ } \{f \le \ell \} of additive planar Gaussian fields are bounded almost surely at the critical level ℓ c = 0 \ell _c = 0 . Since we do not assume positive correlations, this provides the first examples of continuous non-positively-correlated stationary planar Gaussian fields for which the boundedness of the nodal domains has been confirmed. By contrast, in dimension d ≥ 3 d \ge 3 the excursion sets have unbounded components at all levels.
加性高斯场节点域的有界性
本文研究了加性高斯场偏移集的连通性,即其协方差函数分解为分别依赖于坐标的项和的平稳中心高斯场。我们的主要结果是,在温和平滑和相关衰减假设下,加性平面高斯场的偏移集{f≤}α {f \le\ell}在临界能级α c = 0 \ell _c = 0几乎肯定有界。由于我们不假设正相关,这提供了连续非正相关的平稳平面高斯场的第一个例子,其中节点域的有界性已经得到证实。相反,在维度d≥3d \ge 3中,偏移集在所有级别上都具有无界分量。
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 求助全文
来源期刊
CiteScore
1.30
自引率
0.00%
发文量
22
×
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
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
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 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信