{"title":"ASCLT的一些最优条件","authors":"István Berkes, Siegfried Hörmann","doi":"10.1007/s10959-023-01245-w","DOIUrl":null,"url":null,"abstract":"<p><p>Let <math><mrow><msub><mi>X</mi><mn>1</mn></msub><mo>,</mo><msub><mi>X</mi><mn>2</mn></msub><mo>,</mo><mo>…</mo></mrow></math> be independent random variables with <math><mrow><mi>E</mi><msub><mi>X</mi><mi>k</mi></msub><mo>=</mo><mn>0</mn></mrow></math> and <math><mrow><msubsup><mi>σ</mi><mrow><mi>k</mi></mrow><mrow><mspace></mspace><mn>2</mn></mrow></msubsup><mo>:</mo><mo>=</mo><mi>E</mi><msubsup><mi>X</mi><mrow><mi>k</mi></mrow><mn>2</mn></msubsup><mo><</mo><mi>∞</mi></mrow></math> <math><mrow><mo>(</mo><mi>k</mi><mo>≥</mo><mn>1</mn><mo>)</mo></mrow></math>. Set <math><mrow><msub><mi>S</mi><mi>k</mi></msub><mo>=</mo><msub><mi>X</mi><mn>1</mn></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mi>X</mi><mi>k</mi></msub></mrow></math> and assume that <math><mrow><msubsup><mi>s</mi><mrow><mi>k</mi></mrow><mrow><mspace></mspace><mn>2</mn></mrow></msubsup><mo>:</mo><mo>=</mo><mi>E</mi><msubsup><mi>S</mi><mi>k</mi><mn>2</mn></msubsup><mo>→</mo><mi>∞</mi></mrow></math>. We prove that under the Kolmogorov condition <dispformula><math><mrow><mtable><mtr><mtd><mrow><mrow><mo>|</mo></mrow><msub><mi>X</mi><mi>n</mi></msub><mrow><mo>|</mo><mo>≤</mo></mrow><msub><mi>L</mi><mi>n</mi></msub><mo>,</mo><mspace></mspace><msub><mi>L</mi><mi>n</mi></msub><mo>=</mo><mi>o</mi><mrow><mo>(</mo><msub><mi>s</mi><mi>n</mi></msub><mo>/</mo><msup><mrow><mo>(</mo><mo>log</mo><mo>log</mo><msub><mi>s</mi><mi>n</mi></msub><mo>)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>)</mo></mrow></mrow></mtd></mtr></mtable></mrow></math></dispformula>we have <dispformula><math><mrow><mtable><mtr><mtd><mrow><mfrac><mn>1</mn><mrow><mo>log</mo><msubsup><mi>s</mi><mrow><mi>n</mi></mrow><mrow><mspace></mspace><mn>2</mn></mrow></msubsup></mrow></mfrac><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mfrac><msubsup><mi>σ</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow><mrow><mspace></mspace><mn>2</mn></mrow></msubsup><msubsup><mi>s</mi><mrow><mi>k</mi></mrow><mrow><mspace></mspace><mn>2</mn></mrow></msubsup></mfrac><mi>f</mi><mfenced><mfrac><msub><mi>S</mi><mi>k</mi></msub><msub><mi>s</mi><mi>k</mi></msub></mfrac></mfenced><mo>→</mo><mfrac><mn>1</mn><msqrt><mrow><mn>2</mn><mi>π</mi></mrow></msqrt></mfrac><msub><mo>∫</mo><mi>R</mi></msub><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><msup><mi>e</mi><mrow><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup><mo>/</mo><mn>2</mn></mrow></msup><mspace></mspace><mtext>d</mtext><mi>x</mi><mspace></mspace><mrow><mi>a</mi><mo>.</mo><mi>s</mi><mo>.</mo></mrow></mrow></mtd></mtr></mtable></mrow></math></dispformula>for any almost everywhere continuous function <math><mrow><mi>f</mi><mo>:</mo><mi>R</mi><mo>→</mo><mi>R</mi></mrow></math> satisfying <math><mrow><mrow><mo>|</mo><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>|</mo></mrow><mo>≤</mo><msup><mi>e</mi><mrow><mi>γ</mi><msup><mi>x</mi><mn>2</mn></msup></mrow></msup></mrow></math>, <math><mrow><mi>γ</mi><mo><</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow></math>. We also show that replacing the <i>o</i> in (1) by <i>O</i>, relation (2) becomes generally false. Finally, in the case when (1) is not assumed, we give an optimal condition for (2) in terms of the remainder term in the Wiener approximation of the partial sum process <math><mrow><mo>{</mo><msub><mi>S</mi><mi>n</mi></msub><mo>,</mo><mspace></mspace><mi>n</mi><mo>≥</mo><mn>1</mn><mo>}</mo></mrow></math> by a Wiener process.</p>","PeriodicalId":54760,"journal":{"name":"Journal of Theoretical Probability","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10927906/pdf/","citationCount":"0","resultStr":"{\"title\":\"Some Optimal Conditions for the ASCLT.\",\"authors\":\"István Berkes, Siegfried Hörmann\",\"doi\":\"10.1007/s10959-023-01245-w\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p><p>Let <math><mrow><msub><mi>X</mi><mn>1</mn></msub><mo>,</mo><msub><mi>X</mi><mn>2</mn></msub><mo>,</mo><mo>…</mo></mrow></math> be independent random variables with <math><mrow><mi>E</mi><msub><mi>X</mi><mi>k</mi></msub><mo>=</mo><mn>0</mn></mrow></math> and <math><mrow><msubsup><mi>σ</mi><mrow><mi>k</mi></mrow><mrow><mspace></mspace><mn>2</mn></mrow></msubsup><mo>:</mo><mo>=</mo><mi>E</mi><msubsup><mi>X</mi><mrow><mi>k</mi></mrow><mn>2</mn></msubsup><mo><</mo><mi>∞</mi></mrow></math> <math><mrow><mo>(</mo><mi>k</mi><mo>≥</mo><mn>1</mn><mo>)</mo></mrow></math>. Set <math><mrow><msub><mi>S</mi><mi>k</mi></msub><mo>=</mo><msub><mi>X</mi><mn>1</mn></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mi>X</mi><mi>k</mi></msub></mrow></math> and assume that <math><mrow><msubsup><mi>s</mi><mrow><mi>k</mi></mrow><mrow><mspace></mspace><mn>2</mn></mrow></msubsup><mo>:</mo><mo>=</mo><mi>E</mi><msubsup><mi>S</mi><mi>k</mi><mn>2</mn></msubsup><mo>→</mo><mi>∞</mi></mrow></math>. We prove that under the Kolmogorov condition <dispformula><math><mrow><mtable><mtr><mtd><mrow><mrow><mo>|</mo></mrow><msub><mi>X</mi><mi>n</mi></msub><mrow><mo>|</mo><mo>≤</mo></mrow><msub><mi>L</mi><mi>n</mi></msub><mo>,</mo><mspace></mspace><msub><mi>L</mi><mi>n</mi></msub><mo>=</mo><mi>o</mi><mrow><mo>(</mo><msub><mi>s</mi><mi>n</mi></msub><mo>/</mo><msup><mrow><mo>(</mo><mo>log</mo><mo>log</mo><msub><mi>s</mi><mi>n</mi></msub><mo>)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>)</mo></mrow></mrow></mtd></mtr></mtable></mrow></math></dispformula>we have <dispformula><math><mrow><mtable><mtr><mtd><mrow><mfrac><mn>1</mn><mrow><mo>log</mo><msubsup><mi>s</mi><mrow><mi>n</mi></mrow><mrow><mspace></mspace><mn>2</mn></mrow></msubsup></mrow></mfrac><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mfrac><msubsup><mi>σ</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow><mrow><mspace></mspace><mn>2</mn></mrow></msubsup><msubsup><mi>s</mi><mrow><mi>k</mi></mrow><mrow><mspace></mspace><mn>2</mn></mrow></msubsup></mfrac><mi>f</mi><mfenced><mfrac><msub><mi>S</mi><mi>k</mi></msub><msub><mi>s</mi><mi>k</mi></msub></mfrac></mfenced><mo>→</mo><mfrac><mn>1</mn><msqrt><mrow><mn>2</mn><mi>π</mi></mrow></msqrt></mfrac><msub><mo>∫</mo><mi>R</mi></msub><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><msup><mi>e</mi><mrow><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup><mo>/</mo><mn>2</mn></mrow></msup><mspace></mspace><mtext>d</mtext><mi>x</mi><mspace></mspace><mrow><mi>a</mi><mo>.</mo><mi>s</mi><mo>.</mo></mrow></mrow></mtd></mtr></mtable></mrow></math></dispformula>for any almost everywhere continuous function <math><mrow><mi>f</mi><mo>:</mo><mi>R</mi><mo>→</mo><mi>R</mi></mrow></math> satisfying <math><mrow><mrow><mo>|</mo><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>|</mo></mrow><mo>≤</mo><msup><mi>e</mi><mrow><mi>γ</mi><msup><mi>x</mi><mn>2</mn></msup></mrow></msup></mrow></math>, <math><mrow><mi>γ</mi><mo><</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow></math>. We also show that replacing the <i>o</i> in (1) by <i>O</i>, relation (2) becomes generally false. Finally, in the case when (1) is not assumed, we give an optimal condition for (2) in terms of the remainder term in the Wiener approximation of the partial sum process <math><mrow><mo>{</mo><msub><mi>S</mi><mi>n</mi></msub><mo>,</mo><mspace></mspace><mi>n</mi><mo>≥</mo><mn>1</mn><mo>}</mo></mrow></math> by a Wiener process.</p>\",\"PeriodicalId\":54760,\"journal\":{\"name\":\"Journal of Theoretical Probability\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2024-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10927906/pdf/\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Theoretical Probability\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s10959-023-01245-w\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"2023/5/6 0:00:00\",\"PubModel\":\"Epub\",\"JCR\":\"Q3\",\"JCRName\":\"STATISTICS & PROBABILITY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Theoretical Probability","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10959-023-01245-w","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"2023/5/6 0:00:00","PubModel":"Epub","JCR":"Q3","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
Let be independent random variables with and . Set and assume that . We prove that under the Kolmogorov condition we have for any almost everywhere continuous function satisfying , . We also show that replacing the o in (1) by O, relation (2) becomes generally false. Finally, in the case when (1) is not assumed, we give an optimal condition for (2) in terms of the remainder term in the Wiener approximation of the partial sum process by a Wiener process.
期刊介绍:
Journal of Theoretical Probability publishes high-quality, original papers in all areas of probability theory, including probability on semigroups, groups, vector spaces, other abstract structures, and random matrices. This multidisciplinary quarterly provides mathematicians and researchers in physics, engineering, statistics, financial mathematics, and computer science with a peer-reviewed forum for the exchange of vital ideas in the field of theoretical probability.