{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:12:43Z","timestamp":1760145163015,"version":"build-2065373602"},"reference-count":14,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2024,7,1]],"date-time":"2024-07-01T00:00:00Z","timestamp":1719792000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"NSF of Anhui University China","award":["KJ2021A0386"],"award-info":[{"award-number":["KJ2021A0386"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Let {Xn}n\u2208Z be a stationary process with values in a finite set. In this paper, we present a moving average version of the Shannon\u2013McMillan\u2013Breiman theorem; this generalize the corresponding classical results. A sandwich argument reduced the proof to direct applications of the moving strong law of large numbers. The result generalizes the work by Algoet et. al., while relying on a similar sandwich method. It is worth noting that, in some kind of significance, the indices an and \u03d5(n) are symmetrical, i.e., for any integer n, if the growth rate of (an)n\u2208Z is slow enough, all conclusions in this article still hold true.<\/jats:p>","DOI":"10.3390\/sym16070827","type":"journal-article","created":{"date-parts":[[2024,7,3]],"date-time":"2024-07-03T04:23:43Z","timestamp":1719980623000},"page":"827","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On Asymptotic Equipartition Property for Stationary Process of Moving Averages"],"prefix":"10.3390","volume":"16","author":[{"given":"Yuanyuan","family":"Ren","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Xinyang College, Xinyang 464000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhongzhi","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Microelectronics and Data Science, Anhui University of Technology, Ma\u2019anshan 243000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,7,1]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Cover, T.M., and Thomas, J.A. (2005). Elements of Information Theory, Wiley-Interscience. [2nd ed.].","DOI":"10.1002\/047174882X"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"899","DOI":"10.1214\/aop\/1176991794","article-title":"A sandwich proof of the Shannon-McMillan-Breiman theorem","volume":"16","author":"Algoet","year":"1988","journal-title":"Ann. Probab."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"809","DOI":"10.1214\/aoms\/1177706899","article-title":"The Individual Ergodic Theorem of Information Theory","volume":"28","author":"Breiman","year":"1957","journal-title":"Ann. Math. Stat."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"196","DOI":"10.1214\/aoms\/1177729028","article-title":"The basic theorems of information","volume":"24","author":"McMillan","year":"1953","journal-title":"Ann. Math. Stat."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"889","DOI":"10.1017\/S0143385702000457","article-title":"The McMillan Theorem for a Class of Asymptotically Abelian C*-Algebras","volume":"22","author":"Neshveyev","year":"2002","journal-title":"Ergod. Theory Dyn. Syst."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Girardin, V. (2005). On the different extensions of the ergodic theorem of information theory. Recent Advance in Applied Probability, Springer Science+Business Media.","DOI":"10.1007\/0-387-23394-6_7"},{"key":"ref_7","first-page":"265","article-title":"Convergence of averages of point transformations","volume":"49","author":"Akcoglu","year":"1975","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1017\/S0143385700005381","article-title":"Convergence for moving averages","volume":"10","author":"Bellow","year":"1990","journal-title":"Ergod. Theory Dyn. Syst."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1007\/BF01893390","article-title":"Moving averages of ergodic process","volume":"24","author":"Steele","year":"1977","journal-title":"Metrika"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"252","DOI":"10.1090\/S0002-9939-1988-0938678-0","article-title":"Polynomially moving ergodic average","volume":"103","author":"Schwartz","year":"1988","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1007\/s10998-012-7874-5","article-title":"Optimal continued fractions and the moving average ergodic theorem","volume":"66","author":"Haili","year":"2013","journal-title":"Period. Math. Hung."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"761","DOI":"10.1007\/s10959-015-0597-9","article-title":"The generalized entropy ergodicity theorem for nonhomogeneous Markov chains","volume":"29","author":"Wang","year":"2016","journal-title":"J. Theor. Probab."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"13","DOI":"10.1007\/s12044-019-0542-4","article-title":"Markov approximation and the generalized entropy ergodic theorem for non-null stationary process","volume":"130","author":"Wang","year":"2020","journal-title":"Proc. Indian Acad. Sci. (Math. Sci.)"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1367","DOI":"10.1007\/s10959-021-01117-1","article-title":"The generalized entropy ergodicity theorem for nonhomogeneous bifurcating Markov chains indexed by a binary tree","volume":"35","author":"Shi","year":"2022","journal-title":"J. Theor. Probab."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/7\/827\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T15:08:50Z","timestamp":1760108930000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/7\/827"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,7,1]]},"references-count":14,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2024,7]]}},"alternative-id":["sym16070827"],"URL":"https:\/\/doi.org\/10.3390\/sym16070827","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2024,7,1]]}}}