{"title":"A Probabilistic Characterization of Negative Definite Functions.","authors":"Fuchang Gao","doi":"10.1007/978-3-030-26391-1_5","DOIUrl":null,"url":null,"abstract":"<p><p>It is proved that a continuous function <i>f</i> on ℝ <sup><i>n</i></sup> is negative definite if and only if it is polynomially bounded and satisfies the inequality <math><mrow><mi>E</mi> <mi>f</mi> <mo>(</mo> <mi>X</mi> <mo>-</mo> <mi>Y</mi> <mo>)</mo> <mo>≤</mo> <mrow><mi>E</mi> <mi>f</mi> <mo>(</mo> <mi>X</mi> <mo>+</mo> <mi>Y</mi> <mo>)</mo></mrow> </mrow> </math> for all i.i.d. random vectors <i>X</i> and <i>Y</i> in ℝ <sup><i>n</i></sup> . The proof uses Fourier transforms of tempered distributions. The \"only if\" part has been proved earlier by Lifshits et al. (A probabilistic inequality related to negative definite functions.</p>","PeriodicalId":93102,"journal":{"name":"High dimensional probability","volume":"74 ","pages":"41-53"},"PeriodicalIF":0.0000,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7604715/pdf/nihms-1631559.pdf","citationCount":"2","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"High dimensional probability","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/978-3-030-26391-1_5","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"2019/11/27 0:00:00","PubModel":"Epub","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 2
Abstract
It is proved that a continuous function f on ℝ n is negative definite if and only if it is polynomially bounded and satisfies the inequality for all i.i.d. random vectors X and Y in ℝ n . The proof uses Fourier transforms of tempered distributions. The "only if" part has been proved earlier by Lifshits et al. (A probabilistic inequality related to negative definite functions.
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