{"title":"Total variation convergence preserves conditional independence","authors":"Steffen Lauritzen","doi":"10.1016/j.spl.2024.110200","DOIUrl":null,"url":null,"abstract":"<div><p>This note establishes that if a sequence <span><math><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mi>n</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>…</mo></mrow></math></span> of probability measures converges in total variation to the limiting probability measure <span><math><mi>P</mi></math></span>, and <span><math><mi>σ</mi></math></span>-algebras <span><math><mi>A</mi></math></span> and <span><math><mi>B</mi></math></span> are conditionally independent given <span><math><mi>H</mi></math></span> with respect to <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> for all <span><math><mi>n</mi></math></span>, then they are also conditionally independent with respect to the limiting measure <span><math><mi>P</mi></math></span>. As a corollary, this also extends to pointwise convergence of densities to a density.</p></div>","PeriodicalId":49475,"journal":{"name":"Statistics & Probability Letters","volume":"214 ","pages":"Article 110200"},"PeriodicalIF":0.9000,"publicationDate":"2024-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S016771522400169X/pdfft?md5=c56dd1436844d549837b401ab4b369b9&pid=1-s2.0-S016771522400169X-main.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Statistics & Probability Letters","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S016771522400169X","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
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Abstract
This note establishes that if a sequence of probability measures converges in total variation to the limiting probability measure , and -algebras and are conditionally independent given with respect to for all , then they are also conditionally independent with respect to the limiting measure . As a corollary, this also extends to pointwise convergence of densities to a density.
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