平方极值点上的相干分布和渐近线

IF 0.7 4区 数学 Q3 STATISTICS & PROBABILITY
Stanisław Cichomski, Adam Osękowski
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We examine the set <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900224000019_inline9.png\" /> <jats:tex-math> $\\mathrm{ext}(\\mathcal{C})$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> of extreme points of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900224000019_inline10.png\" /> <jats:tex-math> $\\mathcal{C}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and provide its general characterisation. Moreover, we establish several structural properties of finitely-supported elements of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900224000019_inline11.png\" /> <jats:tex-math> $\\mathrm{ext}(\\mathcal{C})$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. We apply these results to obtain the asymptotic sharp bound <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900224000019_inline12.png\" /> <jats:tex-math> $\\lim_{\\alpha \\to \\infty}\\alpha\\cdot(\\sup_{(X,Y)\\in \\mathcal{C}}\\mathbb{E}|X-Y|^{\\alpha}) = {2}/{\\mathrm{e}}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>.","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2024-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Coherent distributions on the square–extreme points and asymptotics\",\"authors\":\"Stanisław Cichomski, Adam Osękowski\",\"doi\":\"10.1017/jpr.2024.1\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0021900224000019_inline1.png\\\" /> <jats:tex-math> $\\\\mathcal{C}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> denote the family of all coherent distributions on the unit square <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0021900224000019_inline2.png\\\" /> <jats:tex-math> $[0,1]^2$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, i.e. all those probability measures <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0021900224000019_inline3.png\\\" /> <jats:tex-math> $\\\\mu$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> for which there exists a random vector <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0021900224000019_inline4.png\\\" /> <jats:tex-math> $(X,Y)\\\\sim \\\\mu$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, a pair <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0021900224000019_inline5.png\\\" /> <jats:tex-math> $(\\\\mathcal{G},\\\\mathcal{H})$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0021900224000019_inline6.png\\\" /> <jats:tex-math> $\\\\sigma$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-fields, and an event <jats:italic>E</jats:italic> such that <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0021900224000019_inline7.png\\\" /> <jats:tex-math> $X=\\\\mathbb{P}(E\\\\mid\\\\mathcal{G})$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0021900224000019_inline8.png\\\" /> <jats:tex-math> $Y=\\\\mathbb{P}(E\\\\mid\\\\mathcal{H})$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> almost surely. We examine the set <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0021900224000019_inline9.png\\\" /> <jats:tex-math> $\\\\mathrm{ext}(\\\\mathcal{C})$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> of extreme points of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0021900224000019_inline10.png\\\" /> <jats:tex-math> $\\\\mathcal{C}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and provide its general characterisation. Moreover, we establish several structural properties of finitely-supported elements of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0021900224000019_inline11.png\\\" /> <jats:tex-math> $\\\\mathrm{ext}(\\\\mathcal{C})$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. We apply these results to obtain the asymptotic sharp bound <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0021900224000019_inline12.png\\\" /> <jats:tex-math> $\\\\lim_{\\\\alpha \\\\to \\\\infty}\\\\alpha\\\\cdot(\\\\sup_{(X,Y)\\\\in \\\\mathcal{C}}\\\\mathbb{E}|X-Y|^{\\\\alpha}) = {2}/{\\\\mathrm{e}}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>.\",\"PeriodicalId\":50256,\"journal\":{\"name\":\"Journal of Applied Probability\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.7000,\"publicationDate\":\"2024-04-05\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Applied Probability\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1017/jpr.2024.1\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"STATISTICS & PROBABILITY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Applied Probability","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1017/jpr.2024.1","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
引用次数: 0

摘要

让 $\mathcal{C}$ 表示单位平方 $[0,1]^2$ 上所有相干分布的族,即存在一个随机向量 $(X,Y)\sim \mu$,一对$(\mathcal{G},\mathcal{H})$ 的$\sigma$ 场,以及一个事件 E,使得 $X=\mathbb{P}(E\mid\mathcal{G})$, $Y=\mathbb{P}(E\mid\mathcal{H})$ 几乎是肯定的。我们研究了 $\mathcal{C}$ 的极值点集合 $\mathrm{ext}(\mathcal{C})$ 并给出了它的一般特征。此外,我们还建立了 $\mathrm{ext}(\mathcal{C})$ 的有限支持元素的几个结构性质。我们应用这些结果得到了渐近尖锐约束 $\lim_{\alpha \to \infty}\alpha\cdot(\sup_{(X,Y)\in \mathcal{C}}\mathbb{E}|X-Y|^{\alpha}) = {2}/\{mathrm{e}}$ 。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Coherent distributions on the square–extreme points and asymptotics
Let $\mathcal{C}$ denote the family of all coherent distributions on the unit square $[0,1]^2$ , i.e. all those probability measures $\mu$ for which there exists a random vector $(X,Y)\sim \mu$ , a pair $(\mathcal{G},\mathcal{H})$ of $\sigma$ -fields, and an event E such that $X=\mathbb{P}(E\mid\mathcal{G})$ , $Y=\mathbb{P}(E\mid\mathcal{H})$ almost surely. We examine the set $\mathrm{ext}(\mathcal{C})$ of extreme points of $\mathcal{C}$ and provide its general characterisation. Moreover, we establish several structural properties of finitely-supported elements of $\mathrm{ext}(\mathcal{C})$ . We apply these results to obtain the asymptotic sharp bound $\lim_{\alpha \to \infty}\alpha\cdot(\sup_{(X,Y)\in \mathcal{C}}\mathbb{E}|X-Y|^{\alpha}) = {2}/{\mathrm{e}}$ .
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来源期刊
Journal of Applied Probability
Journal of Applied Probability 数学-统计学与概率论
CiteScore
1.50
自引率
10.00%
发文量
92
审稿时长
6-12 weeks
期刊介绍: Journal of Applied Probability is the oldest journal devoted to the publication of research in the field of applied probability. It is an international journal published by the Applied Probability Trust, and it serves as a companion publication to the Advances in Applied Probability. Its wide audience includes leading researchers across the entire spectrum of applied probability, including biosciences applications, operations research, telecommunications, computer science, engineering, epidemiology, financial mathematics, the physical and social sciences, and any field where stochastic modeling is used. A submission to Applied Probability represents a submission that may, at the Editor-in-Chief’s discretion, appear in either the Journal of Applied Probability or the Advances in Applied Probability. Typically, shorter papers appear in the Journal, with longer contributions appearing in the Advances.
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