{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:12:00Z","timestamp":1760242320676,"version":"build-2065373602"},"reference-count":19,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2017,5,11]],"date-time":"2017-05-11T00:00:00Z","timestamp":1494460800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>This paper describes and proves two important theorems that compose the Law of Large Numbers for the non-Euclidean     L p    -means, known to be true for the Euclidean     L 2    -means: Let the     L p    -mean estimator, which constitutes the specific functional that estimates the     L p    -mean of N independent and identically distributed random variables; then, (i) the expectation value of the     L p    -mean estimator equals the mean of the distributions of the random variables; and (ii) the limit     N \u2192 \u221e     of the     L p    -mean estimator also equals the mean of the distributions.<\/jats:p>","DOI":"10.3390\/e19050217","type":"journal-article","created":{"date-parts":[[2017,5,11]],"date-time":"2017-05-11T11:06:46Z","timestamp":1494500806000},"page":"217","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["On the Convergence and Law of Large Numbers for the Non-Euclidean Lp -Means"],"prefix":"10.3390","volume":"19","author":[{"given":"George","family":"Livadiotis","sequence":"first","affiliation":[{"name":"Space Science and Engineering, Southwest Research Institute, San Antonio, TX 78238, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2017,5,11]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"518","DOI":"10.1016\/j.physa.2006.09.027","article-title":"Approach to general methods for fitting and their sensitivity","volume":"375","author":"Livadiotis","year":"2007","journal-title":"Physica A"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"2471","DOI":"10.1016\/j.physa.2008.01.002","article-title":"Approach to the block entropy modeling and optimization","volume":"387","author":"Livadiotis","year":"2008","journal-title":"Physica A"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"2375","DOI":"10.3390\/e14122375","article-title":"Expectation value and variance based on \n        \n          \n            \n              L\n              p\n            \n          \n        \n       norms","volume":"14","author":"Livadiotis","year":"2012","journal-title":"Entropy"},{"key":"ref_4","first-page":"436","article-title":"The sunspot as an autonomous dynamical system: A model for the growth and decay phases of sunspots","volume":"379","author":"Livadiotis","year":"2014","journal-title":"J. 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