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Assoc.; citation_volume=99; citation_firstpage=250; citation_lastpage=261; citation_doi=10.1198/016214504000000241"> <meta name="citation_abstract" content="The Dagum family of isotropic covariance functions has two parameters that allow for decoupling of the fractal dimension and the Hurst effect for Gaussian random fields that are stationary and isotropic over Euclidean spaces. Sufficient conditions that allow for positive definiteness in of the Dagum family have been proposed on the basis of the fact that the Dagum family allows for complete monotonicity under some parameter restrictions. The spectral properties of the Dagum family have been inspected to a very limited extent only, and this paper gives insight into this direction. Specifically, we study finite and asymptotic properties of the isotropic spectral density (intended as the Hankel transform) of the Dagum model. Also, we establish some closed-form expressions for the Dagum spectral density in terms of the Fox–Wright functions. Finally, we provide asymptotic properties for such a class of spectral densities."> <meta name="citation_doi" content="10.1017/jpr.2021.103"> <link rel="alternate" href="/core/journals/journal-of-applied-probability/article/abs/deep-look-into-the-dagum-family-of-isotropic-covariance-functions/155F2A83A963782A7792F108B18D9502" hreflang="en" /> <link rel="icon" href="/core/cambridge-core/public/images/favicon.ico" type="image/x-icon"/> <link rel="shortcut icon" href="/core/cambridge-core/public/images/favicon.ico" type="image/x-icon"/> <link href='//fonts.googleapis.com/css?family=Noto+Sans:400,700,400italic,700italic' rel='stylesheet' type='text/css'>  <link rel="stylesheet" href="/core/cambridge-core/public/css/app.css?version=v7.425.0"/> <link rel="dns-prefetch" href="https://usage.prod.aop.cambridge.org/v1/events"> <link rel="stylesheet" href="/core/cambridge-core/public/bower_components/font-awesome/css/font-awesome.min.css"/> <script 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data-v-63dfaf6e>Journal of Applied Probability</span> </a></li><li class="page-breadcrumbs__item" data-v-3692cf84><span aria-hidden="true" class="breadcrumbs-wrapper__arrow" data-v-3692cf84>></span><a href="/core/journals/journal-of-applied-probability/issue/C8B83EE5BAAA7EA926AFE9AF44F35FBC" class="app-link app-link__text app-link--accent" data-v-63dfaf6e data-v-3692cf84><span data-v-63dfaf6e>Volume 59 Issue 4</span> </a></li><li class="page-breadcrumbs__item" data-v-3692cf84><span aria-hidden="true" class="breadcrumbs-wrapper__arrow" data-v-3692cf84>></span><span data-v-3692cf84>A deep look into the dagum family of isotropic covariance...</span></li></ul></div></div> <div class="language" data-v-3692cf84><ul class="language-switch" data-v-6b1118dd data-v-3692cf84><li aria-label="English" data-v-6b1118dd><span class="language-option current divider" data-v-6b1118dd>English</span></li><li aria-label="Français" data-v-6b1118dd><span role="button" tabindex="0" href="#" lang="fr" class="language-option" data-v-6b1118dd> Français </span></li></ul></div></div></div></div> <div class="container container__modified" data-v-01274b1d>  <div class="row" data-v-01274b1d><div role="complementary" aria-label="table of content" class="column__left" data-v-01274b1d><div class="col journal-container row" data-v-146270e8 data-v-01274b1d><img src="https://static.cambridge.org/covers/JPR_0_0_0/journal-of-applied-probability.jpg" alt="" class="journal__image" data-v-146270e8> <a href="/core/journals/journal-of-applied-probability" class="app-link journal__title app-link__text app-link--underlined" data-v-63dfaf6e data-v-146270e8><span class="text" data-v-63dfaf6e>Journal of Applied Probability </span> </a> <hr aria-hidden="true" class="separator default" data-v-7036083a data-v-146270e8></div>  <div id="toc" class="table-of-content" data-v-01274b1d><h2>Article contents</h2> <div id="toc-list-wrapper" class="table-of-content__wrapper"><ul id="toc-list" class="list"><li class="list__item"><a href="#sec0" class="list__item__link"><span class="toc-title">Abstract</span></a></li>  <li class="list__item"><a href="#references-list" class="list__item__link"><span class="toc-title">References</span></a></li></ul></div></div></div> <div class="column__main" data-v-01274b1d><div class="row" data-v-01274b1d><div class="column__main__left" data-v-01274b1d><div id="maincontent" class="col" data-v-862424e6 data-v-01274b1d> <hgroup data-v-862424e6><h1 data-v-862424e6>A deep look into the dagum family of isotropic covariance functions</h1> </hgroup>    <div class="row part-of" data-v-862424e6><span class="part-of__label" data-v-862424e6> Part of: </span> <a href="/core/product/identifier/MSC_2010_STOCHASTIC_PROCESSES/type/BESPOKE_COLLECTION" class="part-of__collection" data-v-f0b31360 data-v-862424e6> Stochastic processes </a><a href="/core/product/identifier/MSC_2010_INTEGRAL_TRANSFORMS_OPERATIONAL_CALCULUS/type/BESPOKE_COLLECTION" class="part-of__collection" data-v-f0b31360 data-v-862424e6> Integral transforms, operational calculus </a></div> <div class="row published-date" data-v-862424e6><p data-v-862424e6> Published online by Cambridge University Press: <strong data-v-862424e6>18 August 2022</strong></p></div>   <div class="contributors-details" data-v-99f6eb26 data-v-862424e6><div class="row contributors" data-v-99f6eb26><div class="col" data-v-99f6eb26><div class="row contributor-type" data-v-792406ce data-v-99f6eb26> <div class="contributor-type__contributor" data-v-792406ce><a href="/core/search?filters%5BauthorTerms%5D=Tarik%20Faouzi&eventCode=SE-AU" class="app-link app-link__text app-link--accent" data-v-63dfaf6e data-v-792406ce><span data-v-63dfaf6e>Tarik Faouzi</span> </a>  <span data-v-792406ce>,</span></div><div class="contributor-type__contributor" data-v-792406ce><a href="/core/search?filters%5BauthorTerms%5D=Emilio%20Porcu&eventCode=SE-AU" class="app-link app-link__text app-link--accent" data-v-63dfaf6e data-v-792406ce><span data-v-63dfaf6e>Emilio Porcu</span> </a>  <span data-v-792406ce>,</span></div><div class="contributor-type__contributor" data-v-792406ce><a href="/core/search?filters%5BauthorTerms%5D=Igor%20Kondrashuk&eventCode=SE-AU" class="app-link app-link__text app-link--accent" data-v-63dfaf6e data-v-792406ce><span data-v-63dfaf6e>Igor Kondrashuk</span> </a> <a target="_blank" href="https://orcid.org/0000-0001-9489-3134" data-test-orcid="Igor Kondrashuk" class="app-link contributor-type__contributor__orcid app-link__icon app-link--" data-v-63dfaf6e data-v-792406ce><img src="data:image/svg+xml;base64,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" alt="Open the ORCID record for Igor Kondrashuk" class="app-icon icon orcid" data-v-d2c09870 data-v-63dfaf6e> <span class="sr-only" data-v-63dfaf6e>[Opens in a new window]</span></a> <span data-v-792406ce> and</span></div><div class="contributor-type__contributor" data-v-792406ce><a href="/core/search?filters%5BauthorTerms%5D=Anatoliy%20Malyarenko&eventCode=SE-AU" class="app-link app-link__text app-link--accent" data-v-63dfaf6e data-v-792406ce><span data-v-63dfaf6e>Anatoliy Malyarenko</span> </a> <a target="_blank" href="https://orcid.org/0000-0002-0139-0747" data-test-orcid="Anatoliy Malyarenko" class="app-link contributor-type__contributor__orcid app-link__icon app-link--" data-v-63dfaf6e data-v-792406ce><img src="data:image/svg+xml;base64,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" alt="Open the ORCID record for Anatoliy Malyarenko" class="app-icon icon orcid" data-v-d2c09870 data-v-63dfaf6e> <span class="sr-only" data-v-63dfaf6e>[Opens in a new window]</span></a> <span data-v-792406ce></span></div> </div> </div> <div class="col-2 collapse-link" data-v-99f6eb26><a href="#authors-details" data-toggle="collapse" aria-expanded="false" aria-controls="authors-details" class="app-link collapsed app-link__text-icon app-link--secondary reverse" data-v-63dfaf6e data-v-99f6eb26><img src="data:image/svg+xml;base64,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" alt="" class="app-icon icon arrow-down" data-v-d2c09870 data-v-63dfaf6e><span class="text" data-v-63dfaf6e>Show author details </span> </a></div></div> <hr aria-hidden="true" class="separator default" data-v-7036083a data-v-99f6eb26> <dl id="authors-details" class="authors-details collapse" data-v-2edb8da6 data-v-99f6eb26><div data-test-author="Tarik Faouzi" class="row author" data-v-2edb8da6><dt class="col-12 col-sm-2 title" data-v-2edb8da6>Tarik Faouzi*</dt> <dd class="col content d-inline d-sm-flex" data-v-2edb8da6><span class="content__title" data-v-2edb8da6>Affiliation:</span> <div class="d-sm-flex flex-column flex-sm-1 d-inline" data-v-2edb8da6><span data-v-2edb8da6><span data-v-2edb8da6>Universidad de Santiago de Chile</span> </span></div></dd></div><div data-test-author="Emilio Porcu" class="row author" data-v-2edb8da6><dt class="col-12 col-sm-2 title" data-v-2edb8da6>Emilio Porcu*</dt> <dd class="col content d-inline d-sm-flex" data-v-2edb8da6><span class="content__title" data-v-2edb8da6>Affiliation:</span> <div class="d-sm-flex flex-column flex-sm-1 d-inline" data-v-2edb8da6><span data-v-2edb8da6><span data-v-2edb8da6>Khalifa University & Trinity College Dublin</span> </span></div></dd></div><div data-test-author="Igor Kondrashuk" class="row author" data-v-2edb8da6><dt class="col-12 col-sm-2 title" data-v-2edb8da6>Igor Kondrashuk*</dt> <dd class="col content d-inline d-sm-flex" data-v-2edb8da6><span class="content__title" data-v-2edb8da6>Affiliation:</span> <div class="d-sm-flex flex-column flex-sm-1 d-inline" data-v-2edb8da6><span data-v-2edb8da6><span data-v-2edb8da6>University of Bio Bio</span> </span></div></dd></div><div data-test-author="Anatoliy Malyarenko" class="row author" data-v-2edb8da6><dt class="col-12 col-sm-2 title" data-v-2edb8da6>Anatoliy Malyarenko*</dt> <dd class="col content d-inline d-sm-flex" data-v-2edb8da6><span class="content__title" data-v-2edb8da6>Affiliation:</span> <div class="d-sm-flex flex-column flex-sm-1 d-inline" data-v-2edb8da6><span data-v-2edb8da6><span data-v-2edb8da6>Mälardalen University</span> </span></div></dd></div> <div class="row" data-v-2edb8da6><dt class="col-sm-2 col-12 title" data-v-2edb8da6> * </dt> <dd class="col content" data-v-2edb8da6><div class="row" data-v-2edb8da6><div class="d-sm-flex d-inline flex-sm-1 flex-sm-wrap" data-v-2edb8da6><div class="d-inline" data-v-2edb8da6><span data-v-2edb8da6><div class="corresp"><span class="label">*</span>Postal address: Departamento de Matemática y Ciencia de la Computación, <span class="institution">Universidad de Santiago de Chile</span>. Email address: <a href="mailto:tarik.faouzi@usach.cl">tarik.faouzi@usach.cl</a></div></span></div><div class="d-inline" data-v-2edb8da6><span data-v-2edb8da6><div class="corresp"><span class="label">**</span>Postal address: Department of Mathematics, <span class="institution">Khalifa University of Science and Technology</span>, <span class="country">Abu Dhabi</span>; School of Computer Science and Statistics, Trinity College Dublin. Email address: <a href="mailto:emilio.porcu@ku.ac.ae">emilio.porcu@ku.ac.ae</a></div></span></div><div class="d-inline" data-v-2edb8da6><span data-v-2edb8da6><div class="corresp"><span class="label">***</span>Postal address: Grupo de Matemática Aplicada & Centro de Ciencias Exactas & Departmento de Ciencias Básicas, <span class="institution">Universidad del Bío-Bío</span>, Campus Fernando May, Av. Andres Bello 720, Casilla 447, Chillán, <span class="country">Chile</span>. Email address: <a href="mailto:igor.kondrashuk@gmail.com">igor.kondrashuk@gmail.com</a></div></span></div><div class="d-inline" data-v-2edb8da6><span data-v-2edb8da6><div class="corresp"><span class="label">****</span>Postal address: Division of Mathematics and Physics, <span class="institution">Mälardalen University</span>, Box 883, 721 23 Västerås, <span class="country">Sweden</span>. Email address: <a href="mailto:anatoliy.malyarenko@mdh.se">anatoliy.malyarenko@mdh.se</a></div></span></div></div></div></dd></div> <hr aria-hidden="true" class="separator default" data-v-7036083a data-v-2edb8da6></dl></div></div> <div id="app-tabs" class="tabs" data-v-1d90c6ce data-v-01274b1d><div id="app-tabs-wrapper" class="tabs__wrapper" data-v-1d90c6ce><div role="navigation" aria-label="tab navigation" class="container" data-v-1d90c6ce><a data-toggle="collapse" href="#appTabs" role="button" aria-expanded="false" aria-controls="appTabs" class="tabs__collapse collapsed d-sm-none d-print-none" data-v-1d90c6ce><span data-v-1d90c6ce></span> <span class="tabs-arrow-up" data-v-1d90c6ce></span> <span class="tabs-arrow-down" data-v-1d90c6ce></span></a> <ul id="appTabs" role="tablist" class="nav nav-tabs tabs__list collapse show" data-v-1d90c6ce><li role="none" class="tabs__tab" data-v-1d90c6ce><a aria-selected="true" href="#article-tab" role="tab" aria-controls="article-tab" tabindex="-1" 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class="toc-title">References</span></a></li></ul></div> <div class="action-bar" data-v-43a4d572><div class="row items"><a href="#access-block" data-test-id="buttonGetAccess" class="get-access-link" data-v-4cc7be75 data-v-43a4d572>Get access</a>  <div class="app-dropdown d-print-none share-dropdown" data-v-fab090b8 data-v-beffd87c data-v-43a4d572><button aria-expanded="false" data-test-id="buttonShareOptions" id="share-dropdown" class="app-button dropdown-menu-button app-button__text-icon app-button--secondary" data-v-2a038744 data-v-fab090b8><img src="/core/page-component/img/share-icon.cbcfad8.svg" alt="" class="app-icon icon share" data-v-d2c09870 data-v-2a038744> <span class="text" data-v-2a038744>Share</span></button> <div aria-labelledby="share-dropdown" class="app-dropdown__menu" style="display:none;" data-v-fab090b8><div aria-labelledby="dropdown-share-button" data-v-fab090b8 data-v-beffd87c> <div class="social-share-container" data-v-fab090b8 data-v-beffd87c></div></div> 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href="https://s100.copyright.com/AppDispatchServlet?publisherName=CUP&publication=JPR&title=A%20deep%20look%20into%20the%20dagum%20family%20of%20isotropic%20covariance%20functions&publicationDate=18%20August%202022&author=Tarik%20Faouzi%2C%20Emilio%20Porcu%2C%20Igor%20Kondrashuk%2C%20Anatoliy%20Malyarenko©right=%C2%A9%20The%20Author(s)%2C%202022.%20Published%20by%20Cambridge%20University%20Press%20on%20behalf%20of%20Applied%20Probability%20Trust&contentID=10.1017%2Fjpr.2021.103&startPage=1026&endPage=1041&orderBeanReset=True&volumeNum=59&issueNum=4&oa=" data-test-id="buttonRightLink" class="app-link rights-link d-print-none app-link__text-icon app-link--accent" data-v-63dfaf6e data-v-92ee52a2 data-v-43a4d572><img src="/core/page-component/img/rights-icon.d4a677c.svg" alt="" class="app-icon icon rights" data-v-d2c09870 data-v-63dfaf6e><span class="text" data-v-63dfaf6e>Rights & Permissions </span> <span class="sr-only" data-v-63dfaf6e>[Opens in a new window]</span></a></div> <hr aria-hidden="true" class="separator default" data-v-7036083a></div> <div class="share-modal-overlay" style="display:none;" data-v-43a4d572></div> <div data-spy="scroll" data-target="#toc" class="scrollspy-content" data-v-43a4d572> <div id="sec0" class="col article-abstract sec" data-v-2fa8b348 data-v-43a4d572><div class="abstract-text-container" data-v-2fa8b348><div lang="en"><h2>Abstract</h2>  <div class="abstract-content"><div class="abstract" data-abstract-type="normal"><p>The Dagum family of isotropic covariance functions has two parameters that allow for decoupling of the fractal dimension and the Hurst effect for Gaussian random fields that are stationary and isotropic over Euclidean spaces. Sufficient conditions that allow for positive definiteness in <span class="inlineFormula"><span class="alternatives"><img class="inline-graphic mathjax-alternative mathjax-alt-graphic mathjax-off" data-mimesubtype="png" data-type src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20221111094610057-0152:S0021900221001030:S0021900221001030_inline1.png" /><span class="mathjax-tex-wrapper" data-mathjax-type="texmath"><span class="tex-math mathjax-tex-math mathjax-on"> $\mathbb{R}^d$ </span></span></span></span> of the Dagum family have been proposed on the basis of the fact that the Dagum family allows for complete monotonicity under some parameter restrictions. The spectral properties of the Dagum family have been inspected to a very limited extent only, and this paper gives insight into this direction. Specifically, we study finite and asymptotic properties of the isotropic spectral density (intended as the Hankel transform) of the Dagum model. Also, we establish some closed-form expressions for the Dagum spectral density in terms of the Fox–Wright functions. Finally, we provide asymptotic properties for such a class of spectral densities.</p></div></div> <hr aria-hidden="true" class="abstract-divider separator default" data-v-7036083a></div></div>   </div>  <div class="keywords" data-v-86c27100 data-v-43a4d572><h2 data-v-86c27100>Keywords</h2> <div class="row keywords__pills" data-v-86c27100><a href="/core/search?filters[keywords]=Hankel transforms" data-v-f0b31360 data-v-86c27100><span data-v-f0b31360 data-v-86c27100>Hankel transforms</span></a><a href="/core/search?filters[keywords]=Mellin–Barnes transforms" data-v-f0b31360 data-v-86c27100><span data-v-f0b31360 data-v-86c27100>Mellin–Barnes transforms</span></a><a href="/core/search?filters[keywords]=spectral theory" data-v-f0b31360 data-v-86c27100><span data-v-f0b31360 data-v-86c27100>spectral theory</span></a><a href="/core/search?filters[keywords]=positive definite" data-v-f0b31360 data-v-86c27100><span data-v-f0b31360 data-v-86c27100>positive definite</span></a></div> <hr aria-hidden="true" class="separator default" data-v-7036083a data-v-86c27100></div> <div class="col classification" data-v-e4b45a1c data-v-43a4d572><div class="classification__type" data-v-e4b45a1c><h2 data-v-e4b45a1c>MSC classification</h2> <div class="row classification__subtype" data-v-e4b45a1c><span class="classification__subtype__title" data-v-e4b45a1c>Primary:</span> <a href="/core/search?filters[classificationCodesByType]=MSC%3B60G60%3BRandom%20fields" data-v-f0b31360 data-v-e4b45a1c> 60G60: Random fields </a></div> <div class="row classification__subtype" data-v-e4b45a1c><span class="classification__subtype__title" data-v-e4b45a1c>Secondary:</span> <a href="/core/search?filters[classificationCodesByType]=MSC%3B44A15%3BSpecial%20transforms%20(Legendre%2C%20Hilbert%2C%20etc.)" data-v-f0b31360 data-v-e4b45a1c> 44A15: Special transforms (Legendre, Hilbert, etc.) </a></div></div>   <hr aria-hidden="true" class="separator default" data-v-7036083a data-v-e4b45a1c></div> <dl class="article-details" data-v-6e32a161 data-v-43a4d572><div class="row" data-v-6e32a161><dt class="col-12 col-sm-3 col-md-2_5 title" data-v-6e32a161> Type </dt> <dd class="col content" data-v-6e32a161>Original Article</dd></div> <div class="row" data-v-6e32a161><dt class="col-12 col-sm-3 col-md-2_5 title" data-v-6e32a161> Information </dt> <dd class="col content" data-v-6e32a161><div class="content__journal" data-v-6e32a161><a href="/core/journals/journal-of-applied-probability" class="app-link app-link__text app-link--underlined" data-v-63dfaf6e data-v-6e32a161><span class="text" data-v-63dfaf6e>Journal of Applied Probability </span> </a> <span data-v-6e32a161> , <a href="/core/journals/journal-of-applied-probability/volume/EEF0E966EEB263C946FA6A84BE7492E8" class="app-link app-link__text app-link--underlined" data-v-63dfaf6e data-v-6e32a161><span class="text" data-v-63dfaf6e>Volume 59 </span> </a></span> <span data-v-6e32a161> , <a href="/core/journals/journal-of-applied-probability/issue/C8B83EE5BAAA7EA926AFE9AF44F35FBC" class="app-link app-link__text app-link--underlined" data-v-63dfaf6e data-v-6e32a161><span class="text" data-v-63dfaf6e>Issue 4 </span> </a></span> <span data-v-6e32a161>, December 2022</span>  <span data-v-6e32a161>, pp. 1026 - 1041</span> </div> <div class="doi-data" data-v-6e32a161><div data-v-6e32a161>DOI: <a target="_blank" href="https://doi.org/10.1017/jpr.2021.103" class="app-link app-link__text app-link--accent" data-v-63dfaf6e data-v-6e32a161><span class="text" data-v-63dfaf6e>https://doi.org/10.1017/jpr.2021.103 </span> <span class="sr-only" data-v-63dfaf6e>[Opens in a new window]</span></a></div> <a data-target="crossmark" aria-label="Check for updates" href="#" class="crossmark-widget" data-v-6e32a161><img src="/core/page-component/img/crossmark-logo.61d5da3.svg" alt="Check for updates" data-v-6e32a161></a></div>  </dd></div>   <div class="row" data-v-6e32a161><dt class="col-12 col-sm-3 col-md-2_5 title" data-v-6e32a161> Copyright </dt> <dd class="col content" data-v-6e32a161><div data-v-6e32a161> © The Author(s), 2022. 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Email address: \u003Ca href=\"mailto:tarik.faouzi@usach.cl\"\u003Etarik.faouzi@usach.cl\u003C\u002Fa\u003E\u003C\u002Fdiv\u003E",isAnonymous:a,fullName:"Tarik Faouzi",searchUrl:"\u002Fcore\u002Fsearch?filters%5BauthorTerms%5D=Tarik%20Faouzi&eventCode=SE-AU",orcidUrl:f},{givenNames:"Emilio",surname:"Porcu",nameStyle:f,affiliations:[{text:"Khalifa University & Trinity College Dublin"}],isCorresponding:c,notes:"\u003Cdiv class=\"corresp\"\u003E\u003Cspan class=\"label\"\u003E**\u003C\u002Fspan\u003EPostal address: Department of Mathematics, \u003Cspan class=\"institution\"\u003EKhalifa University of Science and Technology\u003C\u002Fspan\u003E, \u003Cspan class=\"country\"\u003EAbu Dhabi\u003C\u002Fspan\u003E; School of Computer Science and Statistics, Trinity College Dublin. Email address: \u003Ca href=\"mailto:emilio.porcu@ku.ac.ae\"\u003Eemilio.porcu@ku.ac.ae\u003C\u002Fa\u003E\u003C\u002Fdiv\u003E",isAnonymous:a,fullName:"Emilio Porcu",searchUrl:"\u002Fcore\u002Fsearch?filters%5BauthorTerms%5D=Emilio%20Porcu&eventCode=SE-AU",orcidUrl:f},{givenNames:"Igor",surname:"Kondrashuk",nameStyle:f,affiliations:[{text:"University of Bio Bio"}],isCorresponding:c,notes:"\u003Cdiv class=\"corresp\"\u003E\u003Cspan class=\"label\"\u003E***\u003C\u002Fspan\u003EPostal address: Grupo de Matemática Aplicada & Centro de Ciencias Exactas & Departmento de Ciencias Básicas, \u003Cspan class=\"institution\"\u003EUniversidad del Bío-Bío\u003C\u002Fspan\u003E, Campus Fernando May, Av. Andres Bello 720, Casilla 447, Chillán, \u003Cspan class=\"country\"\u003EChile\u003C\u002Fspan\u003E. Email address: \u003Ca href=\"mailto:igor.kondrashuk@gmail.com\"\u003Eigor.kondrashuk@gmail.com\u003C\u002Fa\u003E\u003C\u002Fdiv\u003E",isAnonymous:a,fullName:"Igor Kondrashuk",searchUrl:"\u002Fcore\u002Fsearch?filters%5BauthorTerms%5D=Igor%20Kondrashuk&eventCode=SE-AU",orcidUrl:"https:\u002F\u002Forcid.org\u002F0000-0001-9489-3134"},{givenNames:"Anatoliy",surname:"Malyarenko",nameStyle:f,affiliations:[{text:"Mälardalen University"}],isCorresponding:c,notes:"\u003Cdiv class=\"corresp\"\u003E\u003Cspan class=\"label\"\u003E****\u003C\u002Fspan\u003EPostal address: Division of Mathematics and Physics, \u003Cspan class=\"institution\"\u003EMälardalen University\u003C\u002Fspan\u003E, Box 883, 721 23 Västerås, \u003Cspan class=\"country\"\u003ESweden\u003C\u002Fspan\u003E. Email address: \u003Ca href=\"mailto:anatoliy.malyarenko@mdh.se\"\u003Eanatoliy.malyarenko@mdh.se\u003C\u002Fa\u003E\u003C\u002Fdiv\u003E",isAnonymous:a,fullName:"Anatoliy Malyarenko",searchUrl:"\u002Fcore\u002Fsearch?filters%5BauthorTerms%5D=Anatoliy%20Malyarenko&eventCode=SE-AU",orcidUrl:"https:\u002F\u002Forcid.org\u002F0000-0002-0139-0747"}]},translators:{contributors:[],label:"Translated by"}},collections:[{link:"\u002Fcore\u002Fproduct\u002Fidentifier\u002FMSC_2010_STOCHASTIC_PROCESSES\u002Ftype\u002FBESPOKE_COLLECTION",id:"MSC_2010_STOCHASTIC_PROCESSES",title:"Stochastic processes"},{link:"\u002Fcore\u002Fproduct\u002Fidentifier\u002FMSC_2010_INTEGRAL_TRANSFORMS_OPERATIONAL_CALCULUS\u002Ftype\u002FBESPOKE_COLLECTION",id:"MSC_2010_INTEGRAL_TRANSFORMS_OPERATIONAL_CALCULUS",title:"Integral transforms, operational calculus"}],publishedDate:"18 August 2022",keywords:[{url:"\u002Fcore\u002Fsearch?filters[keywords]=Hankel transforms",name:"Hankel transforms"},{url:"\u002Fcore\u002Fsearch?filters[keywords]=Mellin–Barnes transforms",name:"Mellin–Barnes transforms"},{url:"\u002Fcore\u002Fsearch?filters[keywords]=spectral theory",name:"spectral theory"},{url:"\u002Fcore\u002Fsearch?filters[keywords]=positive definite",name:"positive definite"}],openPracticeBadges:[],doi:{url:"https:\u002F\u002Fdoi.org\u002F10.1017\u002Fjpr.2021.103",value:"10.1017\u002Fjpr.2021.103"},copyright:{statement:["© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust"],holder:["Applied Probability Trust"],year:[2022]},creativeCommons:f,acceptedManuscript:a,type:"research-article",pageRange:{range:"1026 - 1041",firstPage:"1026",lastPage:"1041"},typeDescription:"Original Article",resultTypes:[],commentsCount:x,topicsAndSubtopics:f},journal:{id:y,title:j,titleSlug:z,mnemonic:"JPR",titleHistory:[],isFirstView:a,journalSlug:z,isCompanion:a,parentCompanionJournalName:j,associatedParentCollection:f,paymentInfo:{prices:{"£":{price:26,sku:k,skuNew:i,currency:"£"},"€":{price:31,sku:k,skuNew:i,currency:"€"},US$:{price:36,sku:k,skuNew:i,currency:"US$"},AU$:{price:51,sku:k,skuNew:i,currency:"AU$"}}},url:A,firstViewUrl:"\u002Fcore\u002Fjournals\u002Fjournal-of-applied-probability\u002Ffirstview",coverUrl:"https:\u002F\u002Fstatic.cambridge.org\u002Fcovers\u002FJPR_0_0_0\u002Fjournal-of-applied-probability.jpg",submitMaterialsUrl:"\u002Fcore\u002Fjournals\u002Fjournal-of-applied-probability\u002Finformation\u002Fauthor-instructions\u002Fsubmitting-your-materials",hasHistory:a,latestTitle:j,latestId:y,hasPastTitle:a,issue:{id:"C8B83EE5BAAA7EA926AFE9AF44F35FBC",number:"4",title:"Issue 4",publishedDate:B,printPublishTimestamp:C,url:D},volume:{id:"EEF0E966EEB263C946FA6A84BE7492E8",number:E,title:"Volume 59",publishedDate:B,printPublishTimestamp:C,url:"\u002Fcore\u002Fjournals\u002Fjournal-of-applied-probability\u002Fvolume\u002FEEF0E966EEB263C946FA6A84BE7492E8"}},abstract:{textAbstracts:[{title:"Abstract",content:"\u003Cdiv class=\"abstract\" data-abstract-type=\"normal\"\u003E\u003Cp\u003EThe Dagum family of isotropic covariance functions has two parameters that allow for decoupling of the fractal dimension and the Hurst effect for Gaussian random fields that are stationary and isotropic over Euclidean spaces. Sufficient conditions that allow for positive definiteness in \u003Cspan class=\"inlineFormula\"\u003E\u003Cspan class=\"alternatives\"\u003E\u003Cimg class=\"inline-graphic mathjax-alternative mathjax-alt-graphic mathjax-off\" data-mimesubtype=\"png\" data-type src=\"https:\u002F\u002Fstatic.cambridge.org\u002Fbinary\u002Fversion\u002Fid\u002Furn:cambridge.org:id:binary:20221111094610057-0152:S0021900221001030:S0021900221001030_inline1.png\" \u002F\u003E\u003Cspan class=\"mathjax-tex-wrapper\" data-mathjax-type=\"texmath\"\u003E\u003Cspan class=\"tex-math mathjax-tex-math mathjax-on\"\u003E\n$\\mathbb{R}^d$\n\u003C\u002Fspan\u003E\u003C\u002Fspan\u003E\u003C\u002Fspan\u003E\u003C\u002Fspan\u003E of the Dagum family have been proposed on the basis of the fact that the Dagum family allows for complete monotonicity under some parameter restrictions. The spectral properties of the Dagum family have been inspected to a very limited extent only, and this paper gives insight into this direction. Specifically, we study finite and asymptotic properties of the isotropic spectral density (intended as the Hankel transform) of the Dagum model. Also, we establish some closed-form expressions for the Dagum spectral density in terms of the Fox–Wright functions. Finally, we provide asymptotic properties for such a class of spectral densities.\u003C\u002Fp\u003E\u003C\u002Fdiv\u003E",lang:F}]},content:{html:b,tableOfContent:[],footnotes:[],fulltextNotes:[],references:[{id:"ref1",displayNumber:"[1]",existInContent:a,content:"\u003Cspan class=\"string-name\"\u003E\u003Cspan class=\"surname\"\u003EAllendes\u003C\u002Fspan\u003E, \u003Cspan class=\"given-names\"\u003EP.\u003C\u002Fspan\u003E\u003C\u002Fspan\u003E, \u003Cspan class=\"string-name\"\u003E\u003Cspan class=\"surname\"\u003EKniehl\u003C\u002Fspan\u003E, \u003Cspan class=\"given-names\"\u003EB. A.\u003C\u002Fspan\u003E\u003C\u002Fspan\u003E, \u003Cspan class=\"string-name\"\u003E\u003Cspan class=\"surname\"\u003EKondrashuk\u003C\u002Fspan\u003E, \u003Cspan class=\"given-names\"\u003EI.\u003C\u002Fspan\u003E\u003C\u002Fspan\u003E, \u003Cspan class=\"string-name\"\u003E\u003Cspan class=\"surname\"\u003ENotte-Cuello\u003C\u002Fspan\u003E, \u003Cspan class=\"given-names\"\u003EE. 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Statist.","Statist. Prob. Lett.","2014","2020","2004","book","Table of Integrals, Series, and Products","Advances and Challenges in Space-Time Modelling of Natural Events","Springer","Interpolation of Spatial Data: Some Theory for Kriging","Correlation Theory of Stationary and Related Random Functions","A deep look into the dagum family of isotropic covariance functions",0,"5AE4897C20187657FD3A685CF371BC52","journal-of-applied-probability","\u002Fcore\u002Fjournals\u002Fjournal-of-applied-probability","December 2022",1669852800000,"\u002Fcore\u002Fjournals\u002Fjournal-of-applied-probability\u002Fissue\u002FC8B83EE5BAAA7EA926AFE9AF44F35FBC","59","en","Solution to Bethe–Salpeter equation via Mellin–Barnes transform","Nuclear Phys. 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Assoc.","10.1198\u002F016214504000000241",3));</script><script src="/core/page-component/ee8029a.js" defer></script><script src="/core/page-component/20147ec.js" defer></script><script src="/core/page-component/66884fb.js" defer></script><script src="/core/page-component/24cf9b0.js" defer></script> </div> </div> <div id="article-new-home-productCitations" class="product-citations-modal reveal-modal medium" data-reveal role="dialog" aria-labelledby="article-new-home-citedByModalHeader"> <div class="header"> <div class="heading_07 margin-bottom" id="article-new-home-citedByModalHeader">Cited by</div> </div> <div class="citation-content"> <div class="row collapse header margin-top productCitations-content"> <ul class="small-12 small-centered columns citations"> <li class="fade-in section-container active"> <a href="#" class="section-button" data-id="article-new-home-crossref-citations"> <div class="circular medium citation"> <img src="https://assets.crossref.org/logo/crossref-logo-100.png" alt="Crossref logo"> <span class="citation-count">4</span> </div> </a> </li> <li class="fade-in section-container"> <a href="#" class="section-button" data-id="article-new-home-scholar-citations"> <div class="circular medium citation"> <img src="https://upload.wikimedia.org/wikipedia/commons/a/a9/Google_Scholar_logo_2015.PNG" alt="Google Scholar logo"> </div> </a> </li> </ul> </div> <div class="row wrapper no-padding-top section-content"> <div class="small-10 small-centered columns"> <section class="content" id="article-new-home-crossref-citations"> <div class="large-12 columns"> <div class="panel margin-top-small citation-container"> <div class="row"> <div class="medium-12 columns"> <div class="print-only print-heading margin-bottom-large">Crossref Citations</div> <div class="row margin-bottom"> <div class="small-1 columns"> <a href="http://www.crossref.org/citedby/index.html" target="_blank"> <img src="https://assets.crossref.org/logo/crossref-logo-100.png" alt="Crossref logo"> </a> </div> <div class="small-11 columns"> <h5 class="heading_10"> This article has been cited by the following publications. 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