广义Orlicz球的Maxwell原理

IF 1.5 Q2 PHYSICS, MATHEMATICAL
S. Johnston, J. Prochno
{"title":"广义Orlicz球的Maxwell原理","authors":"S. Johnston, J. Prochno","doi":"10.1214/22-aihp1298","DOIUrl":null,"url":null,"abstract":"In [A dozen de {F}inetti-style results in search of a theory, Ann. Inst. H. Poincar\\'{e} Probab. Statist. 23(2)(1987), 397--423], Diaconis and Freedman studied low-dimensional projections of random vectors from the Euclidean unit sphere and the simplex in high dimensions, noting that the individual coordinates of these random vectors look like Gaussian and exponential random variables respectively. In subsequent works, Rachev and R\\\"uschendorf and Naor and Romik unified these results by establishing a connection between $\\ell_p^N$ balls and a $p$-generalized Gaussian distribution. In this paper, we study similar questions in a significantly generalized and unifying setting, looking at low-dimensional projections of random vectors uniformly distributed on sets of the form \\[B_{\\phi,t}^N := \\Big\\{(s_1,\\ldots,s_N)\\in\\mathbb{R}^N : \\sum_{ i =1}^N\\phi(s_i)\\leq t N\\Big\\},\\] where $\\phi:\\mathbb{R}\\to [0,\\infty]$ is a potential (including the case of Orlicz functions). Our method is different from both Rachev-R\\\"uschendorf and Naor-Romik, based on a large deviation perspective in the form of quantitative versions of Cram\\'er's theorem and the Gibbs conditioning principle, providing a natural framework beyond the $p$-generalized Gaussian distribution while simultaneously unraveling the role this distribution plays in relation to the geometry of $\\ell_p^N$ balls. We find that there is a critical parameter $t_{\\mathrm{crit}}$ at which there is a phase transition in the behaviour of the projections: for $t > t_{\\mathrm{crit}}$ the coordinates of random points sampled from $B_{\\phi,t}^N$ behave like uniform random variables, but for $t \\leq t_{\\mathrm{crit}}$ the Gibbs conditioning principle comes into play, and here there is a parameter $\\beta_t>0$ (the inverse temperature) such that the coordinates are approximately distributed according to a density proportional to $e^{ -\\beta_t\\phi(s)}$.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":null,"pages":null},"PeriodicalIF":1.5000,"publicationDate":"2020-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"9","resultStr":"{\"title\":\"A Maxwell principle for generalized Orlicz balls\",\"authors\":\"S. Johnston, J. Prochno\",\"doi\":\"10.1214/22-aihp1298\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"In [A dozen de {F}inetti-style results in search of a theory, Ann. Inst. H. Poincar\\\\'{e} Probab. Statist. 23(2)(1987), 397--423], Diaconis and Freedman studied low-dimensional projections of random vectors from the Euclidean unit sphere and the simplex in high dimensions, noting that the individual coordinates of these random vectors look like Gaussian and exponential random variables respectively. In subsequent works, Rachev and R\\\\\\\"uschendorf and Naor and Romik unified these results by establishing a connection between $\\\\ell_p^N$ balls and a $p$-generalized Gaussian distribution. In this paper, we study similar questions in a significantly generalized and unifying setting, looking at low-dimensional projections of random vectors uniformly distributed on sets of the form \\\\[B_{\\\\phi,t}^N := \\\\Big\\\\{(s_1,\\\\ldots,s_N)\\\\in\\\\mathbb{R}^N : \\\\sum_{ i =1}^N\\\\phi(s_i)\\\\leq t N\\\\Big\\\\},\\\\] where $\\\\phi:\\\\mathbb{R}\\\\to [0,\\\\infty]$ is a potential (including the case of Orlicz functions). Our method is different from both Rachev-R\\\\\\\"uschendorf and Naor-Romik, based on a large deviation perspective in the form of quantitative versions of Cram\\\\'er's theorem and the Gibbs conditioning principle, providing a natural framework beyond the $p$-generalized Gaussian distribution while simultaneously unraveling the role this distribution plays in relation to the geometry of $\\\\ell_p^N$ balls. We find that there is a critical parameter $t_{\\\\mathrm{crit}}$ at which there is a phase transition in the behaviour of the projections: for $t > t_{\\\\mathrm{crit}}$ the coordinates of random points sampled from $B_{\\\\phi,t}^N$ behave like uniform random variables, but for $t \\\\leq t_{\\\\mathrm{crit}}$ the Gibbs conditioning principle comes into play, and here there is a parameter $\\\\beta_t>0$ (the inverse temperature) such that the coordinates are approximately distributed according to a density proportional to $e^{ -\\\\beta_t\\\\phi(s)}$.\",\"PeriodicalId\":42884,\"journal\":{\"name\":\"Annales de l Institut Henri Poincare D\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.5000,\"publicationDate\":\"2020-12-21\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"9\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Annales de l Institut Henri Poincare D\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1214/22-aihp1298\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"PHYSICS, MATHEMATICAL\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annales de l Institut Henri Poincare D","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1214/22-aihp1298","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
引用次数: 9

摘要

在《十几个德{菲}内蒂式的结果寻找一个理论》一书中,安。也许吧。Statist. 23(2)(1987), 397—423],Diaconis和Freedman研究了来自欧几里得单位球和高维单纯形的随机向量的低维投影,注意到这些随机向量的单个坐标分别看起来像高斯和指数随机变量。在随后的工作中,Rachev和r schendorf以及Naor和Romik通过建立$\ell_p^N$球与$p$广义高斯分布之间的联系来统一这些结果。在本文中,我们在一个显著推广和统一的设置中研究类似的问题,观察均匀分布在形式为\[B_{\phi,t}^N := \Big\{(s_1,\ldots,s_N)\in\mathbb{R}^N : \sum_{ i =1}^N\phi(s_i)\leq t N\Big\},\]的集合上的随机向量的低维投影,其中$\phi:\mathbb{R}\to [0,\infty]$是一个势(包括Orlicz函数的情况)。我们的方法不同于rachev - r schendorf和Naor-Romik,我们的方法基于一个大偏差的视角,以定量版本的克拉姆萨姆定理和吉布斯条件反射原理的形式,提供了一个超越$p$ -广义高斯分布的自然框架,同时揭示了该分布在$\ell_p^N$球的几何形状中所起的作用。我们发现存在一个临界参数$t_{\mathrm{crit}}$,在该参数处,投影的行为发生相变:对于$t > t_{\mathrm{crit}}$,从$B_{\phi,t}^N$中采样的随机点的坐标表现得像均匀随机变量,但对于$t \leq t_{\mathrm{crit}}$,吉布斯条件反射原理开始发挥作用,这里有一个参数$\beta_t>0$(逆温度),使得坐标根据与$e^{ -\beta_t\phi(s)}$成比例的密度近似分布。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
A Maxwell principle for generalized Orlicz balls
In [A dozen de {F}inetti-style results in search of a theory, Ann. Inst. H. Poincar\'{e} Probab. Statist. 23(2)(1987), 397--423], Diaconis and Freedman studied low-dimensional projections of random vectors from the Euclidean unit sphere and the simplex in high dimensions, noting that the individual coordinates of these random vectors look like Gaussian and exponential random variables respectively. In subsequent works, Rachev and R\"uschendorf and Naor and Romik unified these results by establishing a connection between $\ell_p^N$ balls and a $p$-generalized Gaussian distribution. In this paper, we study similar questions in a significantly generalized and unifying setting, looking at low-dimensional projections of random vectors uniformly distributed on sets of the form \[B_{\phi,t}^N := \Big\{(s_1,\ldots,s_N)\in\mathbb{R}^N : \sum_{ i =1}^N\phi(s_i)\leq t N\Big\},\] where $\phi:\mathbb{R}\to [0,\infty]$ is a potential (including the case of Orlicz functions). Our method is different from both Rachev-R\"uschendorf and Naor-Romik, based on a large deviation perspective in the form of quantitative versions of Cram\'er's theorem and the Gibbs conditioning principle, providing a natural framework beyond the $p$-generalized Gaussian distribution while simultaneously unraveling the role this distribution plays in relation to the geometry of $\ell_p^N$ balls. We find that there is a critical parameter $t_{\mathrm{crit}}$ at which there is a phase transition in the behaviour of the projections: for $t > t_{\mathrm{crit}}$ the coordinates of random points sampled from $B_{\phi,t}^N$ behave like uniform random variables, but for $t \leq t_{\mathrm{crit}}$ the Gibbs conditioning principle comes into play, and here there is a parameter $\beta_t>0$ (the inverse temperature) such that the coordinates are approximately distributed according to a density proportional to $e^{ -\beta_t\phi(s)}$.
求助全文
通过发布文献求助,成功后即可免费获取论文全文。 去求助
来源期刊
CiteScore
2.30
自引率
0.00%
发文量
16
×
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