{"title":"随机积分方法的一致性","authors":"Julian Hofstadler, Daniel Rudolf","doi":"10.1016/j.jco.2023.101740","DOIUrl":null,"url":null,"abstract":"<div><p>We prove that a class of randomized integration methods, including averages based on <span><math><mo>(</mo><mi>t</mi><mo>,</mo><mi>d</mi><mo>)</mo></math></span><span>-sequences, Latin hypercube<span><span> sampling, Frolov points as well as Cranley-Patterson rotations, consistently estimates expectations of integrable functions<span>. Consistency here refers to convergence in mean and/or convergence in probability of the estimator to the integral of interest. Moreover, we suggest median modified methods and show for </span></span>integrands in </span></span><span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span> with <span><math><mi>p</mi><mo>></mo><mn>1</mn></math></span><span> consistency in terms of almost sure convergence.</span></p></div>","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"4","resultStr":"{\"title\":\"Consistency of randomized integration methods\",\"authors\":\"Julian Hofstadler, Daniel Rudolf\",\"doi\":\"10.1016/j.jco.2023.101740\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>We prove that a class of randomized integration methods, including averages based on <span><math><mo>(</mo><mi>t</mi><mo>,</mo><mi>d</mi><mo>)</mo></math></span><span>-sequences, Latin hypercube<span><span> sampling, Frolov points as well as Cranley-Patterson rotations, consistently estimates expectations of integrable functions<span>. Consistency here refers to convergence in mean and/or convergence in probability of the estimator to the integral of interest. Moreover, we suggest median modified methods and show for </span></span>integrands in </span></span><span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span> with <span><math><mi>p</mi><mo>></mo><mn>1</mn></math></span><span> consistency in terms of almost sure convergence.</span></p></div>\",\"PeriodicalId\":1,\"journal\":{\"name\":\"Accounts of Chemical Research\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":16.4000,\"publicationDate\":\"2023-06-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"4\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Accounts of Chemical Research\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0885064X23000092\",\"RegionNum\":1,\"RegionCategory\":\"化学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"CHEMISTRY, MULTIDISCIPLINARY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0885064X23000092","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
We prove that a class of randomized integration methods, including averages based on -sequences, Latin hypercube sampling, Frolov points as well as Cranley-Patterson rotations, consistently estimates expectations of integrable functions. Consistency here refers to convergence in mean and/or convergence in probability of the estimator to the integral of interest. Moreover, we suggest median modified methods and show for integrands in with consistency in terms of almost sure convergence.
期刊介绍：
Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance.
Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.
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