{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,8]],"date-time":"2026-02-08T09:39:47Z","timestamp":1770543587924,"version":"3.49.0"},"reference-count":46,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2023,10,31]],"date-time":"2023-10-31T00:00:00Z","timestamp":1698710400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001729","name":"Swedish Foundation for Strategic Research","doi-asserted-by":"publisher","award":["UKR22-0017"],"award-info":[{"award-number":["UKR22-0017"]}],"id":[{"id":"10.13039\/501100001729","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001729","name":"Swedish Foundation for Strategic Research","doi-asserted-by":"publisher","award":["JPMJCR2115"],"award-info":[{"award-number":["JPMJCR2115"]}],"id":[{"id":"10.13039\/501100001729","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001729","name":"Swedish Foundation for Strategic Research","doi-asserted-by":"publisher","award":["274410"],"award-info":[{"award-number":["274410"]}],"id":[{"id":"10.13039\/501100001729","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002241","name":"Japan Science and Technology Agency CREST","doi-asserted-by":"publisher","award":["UKR22-0017"],"award-info":[{"award-number":["UKR22-0017"]}],"id":[{"id":"10.13039\/501100002241","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002241","name":"Japan Science and Technology Agency CREST","doi-asserted-by":"publisher","award":["JPMJCR2115"],"award-info":[{"award-number":["JPMJCR2115"]}],"id":[{"id":"10.13039\/501100002241","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002241","name":"Japan Science and Technology Agency CREST","doi-asserted-by":"publisher","award":["274410"],"award-info":[{"award-number":["274410"]}],"id":[{"id":"10.13039\/501100002241","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100005416","name":"The Research Council of Norway","doi-asserted-by":"publisher","award":["UKR22-0017"],"award-info":[{"award-number":["UKR22-0017"]}],"id":[{"id":"10.13039\/501100005416","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100005416","name":"The Research Council of Norway","doi-asserted-by":"publisher","award":["JPMJCR2115"],"award-info":[{"award-number":["JPMJCR2115"]}],"id":[{"id":"10.13039\/501100005416","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100005416","name":"The Research Council of Norway","doi-asserted-by":"publisher","award":["274410"],"award-info":[{"award-number":["274410"]}],"id":[{"id":"10.13039\/501100005416","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>We consider five types of entropies for Gaussian distribution: Shannon, R\u00e9nyi, generalized R\u00e9nyi, Tsallis and Sharma\u2013Mittal entropy, establishing their interrelations and their properties as the functions of parameters. Then, we consider fractional Gaussian processes, namely fractional, subfractional, bifractional, multifractional and tempered fractional Brownian motions, and compare the entropies of one-dimensional distributions of these processes.<\/jats:p>","DOI":"10.3390\/axioms12111026","type":"journal-article","created":{"date-parts":[[2023,10,31]],"date-time":"2023-10-31T11:16:27Z","timestamp":1698750987000},"page":"1026","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Properties of Various Entropies of Gaussian Distribution and Comparison of Entropies of Fractional Processes"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0139-0747","authenticated-orcid":false,"given":"Anatoliy","family":"Malyarenko","sequence":"first","affiliation":[{"name":"Division of Mathematics and Physics, M\u00e4lardalen University, 721 23 V\u00e4ster\u00e5s, Sweden"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6877-1800","authenticated-orcid":false,"given":"Yuliya","family":"Mishura","sequence":"additional","affiliation":[{"name":"Division of Mathematics and Physics, M\u00e4lardalen University, 721 23 V\u00e4ster\u00e5s, Sweden"},{"name":"Department of Probability Theory, Statistics and Actuarial Mathematics, Taras Shevchenko National University of Kyiv, 64\/13 Volodymyrska St., 01601 Kyiv, Ukraine"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7208-3130","authenticated-orcid":false,"given":"Kostiantyn","family":"Ralchenko","sequence":"additional","affiliation":[{"name":"Department of Probability Theory, Statistics and Actuarial Mathematics, Taras Shevchenko National University of Kyiv, 64\/13 Volodymyrska St., 01601 Kyiv, Ukraine"}]},{"given":"Yevheniia Anastasiia","family":"Rudyk","sequence":"additional","affiliation":[{"name":"Department of Probability Theory, Statistics and Actuarial Mathematics, Taras Shevchenko National University of Kyiv, 64\/13 Volodymyrska St., 01601 Kyiv, Ukraine"}]}],"member":"1968","published-online":{"date-parts":[[2023,10,31]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1002\/j.1538-7305.1948.tb01338.x","article-title":"A mathematical theory of communication","volume":"27","author":"Shannon","year":"1948","journal-title":"Bell Syst. Tech. J."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Cover, T.M., and Thomas, J.A. (2006). Elements of Information Theory, Wiley.","DOI":"10.1002\/047174882X"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1007\/BF00116251","article-title":"Induction of decision trees","volume":"1","author":"Quinlan","year":"1986","journal-title":"Mach. Learn."},{"key":"ref_4","unstructured":"Pathria, R.K. (2011). Statistical Mechanics, Elsevier."},{"key":"ref_5","unstructured":"Schneier, B. (1996). Applied Cryptography: Protocols, Algorithms, and Source Code in C, Wiley."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1093\/genetics\/97.1.145","article-title":"DNA polymorphism detectable by restriction endonucleases","volume":"97","author":"Nei","year":"1981","journal-title":"Genetics"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1731","DOI":"10.1111\/j.1540-6261.1992.tb04681.x","article-title":"Simple technical trading rules and the stochastic properties of stock returns","volume":"47","author":"Brock","year":"1992","journal-title":"J. Financ."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"130","DOI":"10.1175\/1520-0469(1963)020<0130:DNF>2.0.CO;2","article-title":"Deterministic nonperiodic flow","volume":"20","author":"Lorenz","year":"1963","journal-title":"J. Atmos. Sci."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Wasserman, S., and Faust, K. (1994). Social Network Analysis: Methods and Applications, Cambridge University Press.","DOI":"10.1017\/CBO9780511815478"},{"key":"ref_10","first-page":"141","article-title":"A cross-cultural investigation of the triarchic model of well-being in Turkey and the United States","volume":"37","author":"Mullet","year":"2006","journal-title":"J. Cross-Cult. Psychol."},{"key":"ref_11","unstructured":"Nielsen, M.A., and Chuang, I.L. (2000). Quantum Computation and Quantum Information, Cambridge University Press."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"368","DOI":"10.1007\/s11128-022-03720-5","article-title":"Entanglement witness and linear entropy in an open system influenced by FG noise","volume":"21","author":"Rahman","year":"2022","journal-title":"Quantum Inf. Process."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Hua, A., and Chang, S.L. (2009, January 8\u201311). Effective DDoS Attacks Detection Using Generalized Entropy Metric. Proceedings of the Algorithms and Architectures for Parallel Processing; 9th International Conference, ICA3PP 2009, Taipei, Taiwan.","DOI":"10.1007\/978-3-642-03095-6"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1186","DOI":"10.3390\/e14071186","article-title":"Multivariate multi-scale permutation entropy for complexity analysis of Alzheimer\u2019s disease EEG","volume":"14","author":"Morabito","year":"2012","journal-title":"Entropy"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"265","DOI":"10.1016\/j.bspc.2016.08.022","article-title":"Measuring signal fluctuations in gait rhythm time series of patients with Parkinson\u2019s disease using entropy parameters","volume":"31","author":"Wu","year":"2017","journal-title":"Biomed. Signal Process. Control"},{"key":"ref_16","unstructured":"R\u00e9nyi, A. (July, January 20). On measures of entropy and information. Proceedings of the 4th Berkeley Symposium on Mathematical Statistics and Probability, Vol. I, Berkeley, CA, USA."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1007\/BF01016429","article-title":"Possible generalization of Boltzmann-Gibbs statistics","volume":"52","author":"Tsallis","year":"1988","journal-title":"J. Stat. Phys."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1765","DOI":"10.3390\/e13101765","article-title":"The nonadditive entropy Sq and its applications in physics and elsewhere: Some remarks","volume":"13","author":"Tsallis","year":"2011","journal-title":"Entropy"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"205","DOI":"10.1007\/BF01899728","article-title":"Entropy of type (\u03b1, \u03b2) and other generalized measures in information theory","volume":"22","author":"Sharma","year":"1975","journal-title":"Metrika"},{"key":"ref_20","first-page":"122","article-title":"New non-additive measures of relative information","volume":"2","author":"Sharma","year":"1977","journal-title":"J. Combin. Inform. Syst. Sci."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"032003","DOI":"10.1088\/1751-8113\/45\/3\/032003","article-title":"A closed-form expression for the Sharma\u2013Mittal entropy of exponential families","volume":"45","author":"Nielsen","year":"2012","journal-title":"J. Phys. A"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"387","DOI":"10.15559\/21-VMSTA185","article-title":"Convexity and robustness of the R\u00e9nyi entropy","volume":"8","author":"Buryak","year":"2021","journal-title":"Mod. Stoch. Theory Appl."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Stratonovich, R.L. (2020). Theory of Information and Its Value, Springer.","DOI":"10.1007\/978-3-030-22833-0"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1052","DOI":"10.1007\/s13540-023-00155-2","article-title":"Entropy and alternative entropy functionals of fractional Gaussian noise as the functions of Hurst index","volume":"26","author":"Malyarenko","year":"2023","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_25","first-page":"115","article-title":"Wienersche Spiralen und einige andere interessante Kurven im Hilbertschen Raum","volume":"26","author":"Kolmogorov","year":"1940","journal-title":"Dokl. Acad. Sci. USSR"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"405","DOI":"10.1016\/j.spl.2004.06.035","article-title":"Sub-fractional Brownian motion and its relation to occupation times","volume":"69","author":"Bojdecki","year":"2004","journal-title":"Statist. Probab. Lett."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Mishura, Y., and Zili, M. (2018). Stochastic Analysis of Mixed Fractional Gaussian Processes, Elsevier Ltd.","DOI":"10.1016\/B978-1-78548-245-8.50001-X"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"830","DOI":"10.1016\/j.spa.2005.11.013","article-title":"On bifractional Brownian motion","volume":"116","author":"Russo","year":"2006","journal-title":"Stoch. Process. Appl."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"571","DOI":"10.2307\/3318691","article-title":"An elementary approach to a Girsanov formula and other analytical results on fractional Brownian motions","volume":"5","author":"Norros","year":"1999","journal-title":"Bernoulli"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1214\/EJP.v8-125","article-title":"Fractional Ornstein\u2013Uhlenbeck processes","volume":"8","author":"Cheridito","year":"2003","journal-title":"Electron. J. Probab."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"024115","DOI":"10.1103\/PhysRevE.104.024115","article-title":"Inertia triggers nonergodicity of fractional Brownian motion","volume":"104","author":"Cherstvy","year":"2021","journal-title":"Phys. Rev. E"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"083041","DOI":"10.1088\/1367-2630\/aba390","article-title":"Unexpected crossovers in correlated random-diffusivity processes","volume":"22","author":"Wang","year":"2020","journal-title":"New J. Phys."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"422","DOI":"10.1137\/1010093","article-title":"Fractional Brownian motions, fractional noises and applications","volume":"10","author":"Mandelbrot","year":"1968","journal-title":"SIAM Rev."},{"key":"ref_34","unstructured":"Samorodnitsky, G., and Taqqu, M.S. (1994). Stable Non-Gaussian Random Processes, Chapman & Hall."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Mishura, Y.S. (2008). Stochastic Calculus for Fractional Brownian Motion and Related Processes, Springer-Verlag. Lecture Notes in Mathematics.","DOI":"10.1007\/978-3-540-75873-0"},{"key":"ref_36","unstructured":"Peltier, R.F., and L\u00e9vy V\u00e9hel, J. (1995). Multifractional Brownian Motion: Definition and Preliminary Results, INRIA. [Research Report] RR-2645."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1294","DOI":"10.3150\/10-BEJ261","article-title":"Local time and Tanaka formula for a Volterra-type multifractional Gaussian process","volume":"16","author":"Boufoussi","year":"2010","journal-title":"Bernoulli"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"119","DOI":"10.1090\/S0094-9000-2010-00799-X","article-title":"Properties of the paths of a multifractal Brownian motion","volume":"80","author":"Ralchenko","year":"2010","journal-title":"Theory Probab. Math. Statist."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"19","DOI":"10.4171\/RMI\/217","article-title":"Elliptic Gaussian random processes","volume":"13","author":"Benassi","year":"1997","journal-title":"Rev. Mat. Iberoam."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"200","DOI":"10.1016\/j.spa.2005.09.007","article-title":"How rich is the class of multifractional Brownian motions?","volume":"116","author":"Stoev","year":"2006","journal-title":"Stoch. Process. Appl."},{"key":"ref_41","unstructured":"Ayache, A., Cohen, S., and L\u00e9vy V\u00e9hel, J. (2000, January 5\u20139). The covariance structure of multifractional Brownian motion, with application to long range dependence. Proceedings of the 2000 IEEE International Conference on Acoustics, Speech, and Signal Processing. Proceedings (Cat. No. 00CH37100), Istanbul, Turkey."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"2269","DOI":"10.1016\/j.spl.2013.06.016","article-title":"Tempered fractional Brownian motion","volume":"83","author":"Meerschaert","year":"2013","journal-title":"Statist. Probab. Lett."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1016\/j.spl.2017.08.015","article-title":"Tempered fractional Brownian and stable motions of second kind","volume":"132","author":"Sabzikar","year":"2018","journal-title":"Statist. Probab. Lett."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"084310","DOI":"10.1063\/1.4959533","article-title":"Bifurcation dynamics of the tempered fractional Langevin equation","volume":"26","author":"Zeng","year":"2016","journal-title":"Chaos"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"484","DOI":"10.1007\/s10959-020-01068-z","article-title":"How does tempering affect the local and global properties of fractional Brownian motion?","volume":"35","author":"Azmoodeh","year":"2022","journal-title":"J. Theoret. Probab."},{"key":"ref_46","unstructured":"Gradshteyn, I.S., and Ryzhik, I.M. (2020). Table of Integrals, Series, and Products, Academic Press."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/11\/1026\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T21:15:09Z","timestamp":1760130909000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/11\/1026"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,10,31]]},"references-count":46,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2023,11]]}},"alternative-id":["axioms12111026"],"URL":"https:\/\/doi.org\/10.3390\/axioms12111026","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,10,31]]}}}