{"title":"On the Euclidean Distance between Two Gaussian Points and the Normal Covariogram of $$\\boldsymbol{\\mathbb{R}}^{\\boldsymbol{d}}$$","authors":"D. M. Martirosyan, V. K. Ohanyan","doi":"10.3103/s1068362324010059","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>The concept of covariogram is extended from bounded convex bodies in <span>\\(\\mathbb{R}^{d}\\)</span> to the entire space <span>\\(\\mathbb{R}^{d}\\)</span> by obtaining integral representations for the distribution and probability density functions of the Euclidean distance between two <span>\\(d\\)</span>-dimensional Gaussian points that have correlated coordinates governed by a covariance matrix. When <span>\\(d=2\\)</span>, a closed-form expression for the density function is obtained. Precise bounds for the moments of the considered distance are found in terms of the extreme eigenvalues of the covariance matrix.</p>","PeriodicalId":54854,"journal":{"name":"Journal of Contemporary Mathematical Analysis-Armenian Academy of Sciences","volume":"6 1","pages":""},"PeriodicalIF":0.3000,"publicationDate":"2024-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Contemporary Mathematical Analysis-Armenian Academy of Sciences","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.3103/s1068362324010059","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
The concept of covariogram is extended from bounded convex bodies in \(\mathbb{R}^{d}\) to the entire space \(\mathbb{R}^{d}\) by obtaining integral representations for the distribution and probability density functions of the Euclidean distance between two \(d\)-dimensional Gaussian points that have correlated coordinates governed by a covariance matrix. When \(d=2\), a closed-form expression for the density function is obtained. Precise bounds for the moments of the considered distance are found in terms of the extreme eigenvalues of the covariance matrix.
期刊介绍：
Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences) is an outlet for research stemming from the widely acclaimed Armenian school of theory of functions, this journal today continues the traditions of that school in the area of general analysis. A very prolific group of mathematicians in Yerevan contribute to this leading mathematics journal in the following fields: real and complex analysis; approximations; boundary value problems; integral and stochastic geometry; differential equations; probability; integral equations; algebra.
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