{"title":"Rd 中高斯分布的经验近似值","authors":"Daniel Bartl , Shahar Mendelson","doi":"10.1016/j.aim.2024.110041","DOIUrl":null,"url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span> be independent copies of the standard gaussian random vector in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>. We show that there is an absolute constant <em>c</em> such that for any <span><math><mi>A</mi><mo>⊂</mo><msup><mrow><mi>S</mi></mrow><mrow><mi>d</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span>, with probability at least <span><math><mn>1</mn><mo>−</mo><mn>2</mn><mi>exp</mi><mo></mo><mo>(</mo><mo>−</mo><mi>c</mi><mi>Δ</mi><mi>m</mi><mo>)</mo></math></span>, for every <span><math><mi>t</mi><mo>∈</mo><mi>R</mi></math></span>,<span><span><span><math><munder><mi>sup</mi><mrow><mi>x</mi><mo>∈</mo><mi>A</mi></mrow></munder><mo></mo><mrow><mo>|</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>m</mi></mrow></mfrac><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>m</mi></mrow></munderover><msub><mrow><mn>1</mn></mrow><mrow><mo>{</mo><mo>〈</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><mi>x</mi><mo>〉</mo><mo>≤</mo><mi>t</mi><mo>}</mo></mrow></msub><mo>−</mo><mi>P</mi><mo>(</mo><mo>〈</mo><mi>G</mi><mo>,</mo><mi>x</mi><mo>〉</mo><mo>≤</mo><mi>t</mi><mo>)</mo><mo>|</mo></mrow><mo>≤</mo><mi>Δ</mi><mo>+</mo><mi>σ</mi><mo>(</mo><mi>t</mi><mo>)</mo><msqrt><mrow><mi>Δ</mi></mrow></msqrt><mo>.</mo></math></span></span></span> Here <span><math><mi>σ</mi><mo>(</mo><mi>t</mi><mo>)</mo></math></span> is the variance of <span><math><msub><mrow><mn>1</mn></mrow><mrow><mo>{</mo><mo>〈</mo><mi>G</mi><mo>,</mo><mi>x</mi><mo>〉</mo><mo>≤</mo><mi>t</mi><mo>}</mo></mrow></msub></math></span> and <span><math><mi>Δ</mi><mo>≥</mo><msub><mrow><mi>Δ</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>, where <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> is determined by an unexpected complexity parameter of <em>A</em> that captures the set's geometry (Talagrand's <span><math><msub><mrow><mi>γ</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> functional). The bound, the probability estimate, and the value of <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> are all (almost) optimal.</div><div>We use this fact to show that if <span><math><mi>Γ</mi><mo>=</mo><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>m</mi></mrow></msubsup><mo>〈</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><mi>x</mi><mo>〉</mo><msub><mrow><mi>e</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> is the random matrix that has <span><math><msub><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span> as its rows, then the structure of <span><math><mi>Γ</mi><mo>(</mo><mi>A</mi><mo>)</mo><mo>=</mo><mo>{</mo><mi>Γ</mi><mi>x</mi><mo>:</mo><mi>x</mi><mo>∈</mo><mi>A</mi><mo>}</mo></math></span> is far more rigid and well-prescribed than was previously expected.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"460 ","pages":"Article 110041"},"PeriodicalIF":1.5000,"publicationDate":"2024-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Empirical approximation of the gaussian distribution in Rd\",\"authors\":\"Daniel Bartl , Shahar Mendelson\",\"doi\":\"10.1016/j.aim.2024.110041\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>Let <span><math><msub><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span> be independent copies of the standard gaussian random vector in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>. We show that there is an absolute constant <em>c</em> such that for any <span><math><mi>A</mi><mo>⊂</mo><msup><mrow><mi>S</mi></mrow><mrow><mi>d</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span>, with probability at least <span><math><mn>1</mn><mo>−</mo><mn>2</mn><mi>exp</mi><mo></mo><mo>(</mo><mo>−</mo><mi>c</mi><mi>Δ</mi><mi>m</mi><mo>)</mo></math></span>, for every <span><math><mi>t</mi><mo>∈</mo><mi>R</mi></math></span>,<span><span><span><math><munder><mi>sup</mi><mrow><mi>x</mi><mo>∈</mo><mi>A</mi></mrow></munder><mo></mo><mrow><mo>|</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>m</mi></mrow></mfrac><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>m</mi></mrow></munderover><msub><mrow><mn>1</mn></mrow><mrow><mo>{</mo><mo>〈</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><mi>x</mi><mo>〉</mo><mo>≤</mo><mi>t</mi><mo>}</mo></mrow></msub><mo>−</mo><mi>P</mi><mo>(</mo><mo>〈</mo><mi>G</mi><mo>,</mo><mi>x</mi><mo>〉</mo><mo>≤</mo><mi>t</mi><mo>)</mo><mo>|</mo></mrow><mo>≤</mo><mi>Δ</mi><mo>+</mo><mi>σ</mi><mo>(</mo><mi>t</mi><mo>)</mo><msqrt><mrow><mi>Δ</mi></mrow></msqrt><mo>.</mo></math></span></span></span> Here <span><math><mi>σ</mi><mo>(</mo><mi>t</mi><mo>)</mo></math></span> is the variance of <span><math><msub><mrow><mn>1</mn></mrow><mrow><mo>{</mo><mo>〈</mo><mi>G</mi><mo>,</mo><mi>x</mi><mo>〉</mo><mo>≤</mo><mi>t</mi><mo>}</mo></mrow></msub></math></span> and <span><math><mi>Δ</mi><mo>≥</mo><msub><mrow><mi>Δ</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>, where <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> is determined by an unexpected complexity parameter of <em>A</em> that captures the set's geometry (Talagrand's <span><math><msub><mrow><mi>γ</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> functional). The bound, the probability estimate, and the value of <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> are all (almost) optimal.</div><div>We use this fact to show that if <span><math><mi>Γ</mi><mo>=</mo><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>m</mi></mrow></msubsup><mo>〈</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><mi>x</mi><mo>〉</mo><msub><mrow><mi>e</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> is the random matrix that has <span><math><msub><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span> as its rows, then the structure of <span><math><mi>Γ</mi><mo>(</mo><mi>A</mi><mo>)</mo><mo>=</mo><mo>{</mo><mi>Γ</mi><mi>x</mi><mo>:</mo><mi>x</mi><mo>∈</mo><mi>A</mi><mo>}</mo></math></span> is far more rigid and well-prescribed than was previously expected.</div></div>\",\"PeriodicalId\":50860,\"journal\":{\"name\":\"Advances in Mathematics\",\"volume\":\"460 \",\"pages\":\"Article 110041\"},\"PeriodicalIF\":1.5000,\"publicationDate\":\"2024-11-26\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Advances in Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0001870824005577\",\"RegionNum\":1,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Advances in Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0001870824005577","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
Empirical approximation of the gaussian distribution in Rd
Let be independent copies of the standard gaussian random vector in . We show that there is an absolute constant c such that for any , with probability at least , for every , Here is the variance of and , where is determined by an unexpected complexity parameter of A that captures the set's geometry (Talagrand's functional). The bound, the probability estimate, and the value of are all (almost) optimal.
We use this fact to show that if is the random matrix that has as its rows, then the structure of is far more rigid and well-prescribed than was previously expected.
期刊介绍:
Emphasizing contributions that represent significant advances in all areas of pure mathematics, Advances in Mathematics provides research mathematicians with an effective medium for communicating important recent developments in their areas of specialization to colleagues and to scientists in related disciplines.