{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:00:53Z","timestamp":1760238053588,"version":"build-2065373602"},"reference-count":30,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2022,7,2]],"date-time":"2022-07-02T00:00:00Z","timestamp":1656720000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"German Research Foundation (Deutsche Forschungsgemeinschaft)","award":["EXC 2050\/1\u2014Project ID 390696704"],"award-info":[{"award-number":["EXC 2050\/1\u2014Project ID 390696704"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Based on the canonical correlation analysis, we derive series representations of the probability density function (PDF) and the cumulative distribution function (CDF) of the information density of arbitrary Gaussian random vectors as well as a general formula to calculate the central moments. Using the general results, we give closed-form expressions of the PDF and CDF and explicit formulas of the central moments for important special cases. Furthermore, we derive recurrence formulas and tight approximations of the general series representations, which allow efficient numerical calculations with an arbitrarily high accuracy as demonstrated with an implementation in Python publicly available on GitLab. Finally, we discuss the (in)validity of Gaussian approximations of the information density.<\/jats:p>","DOI":"10.3390\/e24070924","type":"journal-article","created":{"date-parts":[[2022,7,2]],"date-time":"2022-07-02T11:12:35Z","timestamp":1656760355000},"page":"924","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On the Distribution of the Information Density of Gaussian Random Vectors: Explicit Formulas and Tight Approximations"],"prefix":"10.3390","volume":"24","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2465-7181","authenticated-orcid":false,"given":"Jonathan E. W.","family":"Huffmann","sequence":"first","affiliation":[{"name":"Lehrstuhl f\u00fcr Theoretische Informationstechnik, Fakult\u00e4t f\u00fcr Elektrotechnik und Informationstechnik, Technische Universit\u00e4t M\u00fcnchen, 80290 M\u00fcnchen, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6219-6967","authenticated-orcid":false,"given":"Martin","family":"Mittelbach","sequence":"additional","affiliation":[{"name":"Lehrstuhl f\u00fcr Theoretische Nachrichtentechnik, Fakult\u00e4t f\u00fcr Elektrotechnik und Informationstechnik, Technische Universit\u00e4t Dresden, 01062 Dresden, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,7,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"752","DOI":"10.1109\/18.256486","article-title":"Approximation Theory of Output Statistics","volume":"39","author":"Han","year":"1993","journal-title":"IEEE Trans. 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