{"id":"https://openalex.org/W2149619889","doi":"https://doi.org/10.1109/tit.2011.2133710","title":"Computation and Estimation of Generalized Entropy Rates for Denumerable Markov Chains","display_name":"Computation and Estimation of Generalized Entropy Rates for Denumerable Markov Chains","publication_year":2011,"publication_date":"2011-06-21","ids":{"openalex":"https://openalex.org/W2149619889","doi":"https://doi.org/10.1109/tit.2011.2133710","mag":"2149619889"},"language":"en","primary_location":{"id":"doi:10.1109/tit.2011.2133710","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tit.2011.2133710","pdf_url":null,"source":{"id":"https://openalex.org/S4502562","display_name":"IEEE Transactions on Information Theory","issn_l":"0018-9448","issn":["0018-9448","1557-9654"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Information Theory","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://hal.science/hal-01082088","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5050838200","display_name":"Gabriela Ciuperca","orcid":"https://orcid.org/0000-0001-5467-9774"},"institutions":[{"id":"https://openalex.org/I100532134","display_name":"Universit\u00e9 Claude Bernard Lyon 1","ror":"https://ror.org/029brtt94","country_code":"FR","type":"education","lineage":["https://openalex.org/I100532134","https://openalex.org/I203339264"]}],"countries":["FR"],"is_corresponding":true,"raw_author_name":"Gabriela Ciuperca","raw_affiliation_strings":["Laboratoire de Probabilit\u00e9, Combinatoire et Statistique, Universit\u00e9 de Lyon, Villeurbanne, France","LAPCS - Laboratoire de Probabilit\u00e9s, Combinatoire et Statistique (France)"],"affiliations":[{"raw_affiliation_string":"Laboratoire de Probabilit\u00e9, Combinatoire et Statistique, Universit\u00e9 de Lyon, Villeurbanne, France","institution_ids":["https://openalex.org/I100532134"]},{"raw_affiliation_string":"LAPCS - Laboratoire de Probabilit\u00e9s, Combinatoire et Statistique (France)","institution_ids":[]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5060675384","display_name":"Val\u00e9rie Girardin","orcid":"https://orcid.org/0000-0001-9934-3561"},"institutions":[{"id":"https://openalex.org/I4210155549","display_name":"Laboratoire de Math\u00e9matiques Blaise Pascal","ror":"https://ror.org/05sd5r855","country_code":"FR","type":"facility","lineage":["https://openalex.org/I1294671590","https://openalex.org/I1294671590","https://openalex.org/I198244214","https://openalex.org/I4210141950","https://openalex.org/I4210155549"]},{"id":"https://openalex.org/I98702875","display_name":"Universit\u00e9 de Caen Normandie","ror":"https://ror.org/051kpcy16","country_code":"FR","type":"education","lineage":["https://openalex.org/I4210105918","https://openalex.org/I98702875"]},{"id":"https://openalex.org/I4387155825","display_name":"Laboratoire de Math\u00e9matiques Nicolas Oresme","ror":"https://ror.org/03jm2hc44","country_code":null,"type":"facility","lineage":["https://openalex.org/I1294671590","https://openalex.org/I4210105918","https://openalex.org/I4387155825","https://openalex.org/I98702875"]}],"countries":["FR"],"is_corresponding":false,"raw_author_name":"Valerie Girardin","raw_affiliation_strings":["Laboratoire de Math\u00e9matiques N. Oresme, UMR6139, Campus II, BP5186, Universit\u00e9 de Caen Basse Normandie, Caen, France","LMNO - Laboratoire de Math\u00e9matiques Nicolas Oresme (Boulevard du Mar\u00e9chal Juin 14032 CAEN CEDEX 5 - France)"],"affiliations":[{"raw_affiliation_string":"Laboratoire de Math\u00e9matiques N. Oresme, UMR6139, Campus II, BP5186, Universit\u00e9 de Caen Basse Normandie, Caen, France","institution_ids":["https://openalex.org/I98702875"]},{"raw_affiliation_string":"LMNO - Laboratoire de Math\u00e9matiques Nicolas Oresme (Boulevard du Mar\u00e9chal Juin 14032 CAEN CEDEX 5 - France)","institution_ids":["https://openalex.org/I4210155549","https://openalex.org/I4387155825"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5015311118","display_name":"Lo\u00efck Lhote","orcid":null},"institutions":[{"id":"https://openalex.org/I4210139483","display_name":"GREYC","ror":"https://ror.org/043749971","country_code":"FR","type":"facility","lineage":["https://openalex.org/I1294671590","https://openalex.org/I1294671590","https://openalex.org/I194210350","https://openalex.org/I4210105918","https://openalex.org/I4210105918","https://openalex.org/I4210139483","https://openalex.org/I4210159245","https://openalex.org/I98702875"]},{"id":"https://openalex.org/I98702875","display_name":"Universit\u00e9 de Caen Normandie","ror":"https://ror.org/051kpcy16","country_code":"FR","type":"education","lineage":["https://openalex.org/I4210105918","https://openalex.org/I98702875"]}],"countries":["FR"],"is_corresponding":false,"raw_author_name":"Lo\u00efck Lhote","raw_affiliation_strings":["ENSICAEN, GREYC, Campus II, BP5186, Universit\u00e9 de Caen Basse Normandie, Caen, France","Equipe AMACC - Laboratoire GREYC - UMR6072 (France)"],"affiliations":[{"raw_affiliation_string":"ENSICAEN, GREYC, Campus II, BP5186, Universit\u00e9 de Caen Basse Normandie, Caen, France","institution_ids":["https://openalex.org/I98702875"]},{"raw_affiliation_string":"Equipe AMACC - Laboratoire GREYC - UMR6072 (France)","institution_ids":["https://openalex.org/I4210139483"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5050838200"],"corresponding_institution_ids":["https://openalex.org/I100532134"],"apc_list":null,"apc_paid":null,"fwci":0.2446,"has_fulltext":false,"cited_by_count":31,"citation_normalized_percentile":{"value":0.60135173,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"57","issue":"7","first_page":"4026","last_page":"4034"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12261","display_name":"Statistical Mechanics and Entropy","score":0.9997000098228455,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T12261","display_name":"Statistical Mechanics and Entropy","score":0.9997000098228455,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12814","display_name":"Gaussian Processes and Bayesian Inference","score":0.9868999719619751,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11270","display_name":"Complex Systems and Time Series Analysis","score":0.9821000099182129,"subfield":{"id":"https://openalex.org/subfields/2002","display_name":"Economics and Econometrics"},"field":{"id":"https://openalex.org/fields/20","display_name":"Economics, Econometrics and Finance"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8012721538543701},{"id":"https://openalex.org/keywords/entropy-rate","display_name":"Entropy rate","score":0.7508460283279419},{"id":"https://openalex.org/keywords/min-entropy","display_name":"Min entropy","score":0.5756341218948364},{"id":"https://openalex.org/keywords/maximum-entropy-probability-distribution","display_name":"Maximum entropy probability distribution","score":0.5544044971466064},{"id":"https://openalex.org/keywords/markov-chain","display_name":"Markov chain","score":0.5285177230834961},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.5114030241966248},{"id":"https://openalex.org/keywords/entropy","display_name":"Entropy (arrow of time)","score":0.4783075153827667},{"id":"https://openalex.org/keywords/r\u00e9nyi-entropy","display_name":"R\u00e9nyi entropy","score":0.4647729992866516},{"id":"https://openalex.org/keywords/entropy-power-inequality","display_name":"Entropy power inequality","score":0.44965317845344543},{"id":"https://openalex.org/keywords/estimator","display_name":"Estimator","score":0.4465336203575134},{"id":"https://openalex.org/keywords/statistical-physics","display_name":"Statistical physics","score":0.4413774013519287},{"id":"https://openalex.org/keywords/joint-quantum-entropy","display_name":"Joint quantum entropy","score":0.44023072719573975},{"id":"https://openalex.org/keywords/joint-entropy","display_name":"Joint entropy","score":0.4316948652267456},{"id":"https://openalex.org/keywords/markov-process","display_name":"Markov process","score":0.4270251989364624},{"id":"https://openalex.org/keywords/conditional-entropy","display_name":"Conditional entropy","score":0.416690468788147},{"id":"https://openalex.org/keywords/principle-of-maximum-entropy","display_name":"Principle of maximum entropy","score":0.3334994316101074},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.22726327180862427}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8012721538543701},{"id":"https://openalex.org/C125252325","wikidata":"https://www.wikidata.org/wiki/Q1345213","display_name":"Entropy rate","level":4,"score":0.7508460283279419},{"id":"https://openalex.org/C196083917","wikidata":"https://www.wikidata.org/wiki/Q17103874","display_name":"Min entropy","level":5,"score":0.5756341218948364},{"id":"https://openalex.org/C60507348","wikidata":"https://www.wikidata.org/wiki/Q6795892","display_name":"Maximum entropy probability distribution","level":3,"score":0.5544044971466064},{"id":"https://openalex.org/C98763669","wikidata":"https://www.wikidata.org/wiki/Q176645","display_name":"Markov chain","level":2,"score":0.5285177230834961},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5114030241966248},{"id":"https://openalex.org/C106301342","wikidata":"https://www.wikidata.org/wiki/Q4117933","display_name":"Entropy (arrow of time)","level":2,"score":0.4783075153827667},{"id":"https://openalex.org/C142611142","wikidata":"https://www.wikidata.org/wiki/Q1433083","display_name":"R\u00e9nyi entropy","level":3,"score":0.4647729992866516},{"id":"https://openalex.org/C320580","wikidata":"https://www.wikidata.org/wiki/Q5380810","display_name":"Entropy power inequality","level":5,"score":0.44965317845344543},{"id":"https://openalex.org/C185429906","wikidata":"https://www.wikidata.org/wiki/Q1130160","display_name":"Estimator","level":2,"score":0.4465336203575134},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.4413774013519287},{"id":"https://openalex.org/C42047476","wikidata":"https://www.wikidata.org/wiki/Q6269544","display_name":"Joint quantum entropy","level":3,"score":0.44023072719573975},{"id":"https://openalex.org/C106752470","wikidata":"https://www.wikidata.org/wiki/Q1364826","display_name":"Joint entropy","level":3,"score":0.4316948652267456},{"id":"https://openalex.org/C159886148","wikidata":"https://www.wikidata.org/wiki/Q176645","display_name":"Markov process","level":2,"score":0.4270251989364624},{"id":"https://openalex.org/C101721835","wikidata":"https://www.wikidata.org/wiki/Q813908","display_name":"Conditional entropy","level":3,"score":0.416690468788147},{"id":"https://openalex.org/C9679016","wikidata":"https://www.wikidata.org/wiki/Q1417473","display_name":"Principle of maximum entropy","level":2,"score":0.3334994316101074},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.22726327180862427},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1109/tit.2011.2133710","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tit.2011.2133710","pdf_url":null,"source":{"id":"https://openalex.org/S4502562","display_name":"IEEE Transactions on Information Theory","issn_l":"0018-9448","issn":["0018-9448","1557-9654"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Information Theory","raw_type":"journal-article"},{"id":"pmh:oai:HAL:hal-01082088v1","is_oa":true,"landing_page_url":"https://hal.science/hal-01082088","pdf_url":null,"source":{"id":"https://openalex.org/S4306402512","display_name":"HAL (Le Centre pour la Communication Scientifique Directe)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1294671590","host_organization_name":"Centre National de la Recherche Scientifique","host_organization_lineage":["https://openalex.org/I1294671590"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"IEEE Transactions on Information Theory, 2011, 57, pp.4026 - 4034. &#x27E8;10.1109/TIT.2011.2133710&#x27E9;","raw_type":"Journal articles"}],"best_oa_location":{"id":"pmh:oai:HAL:hal-01082088v1","is_oa":true,"landing_page_url":"https://hal.science/hal-01082088","pdf_url":null,"source":{"id":"https://openalex.org/S4306402512","display_name":"HAL (Le Centre pour la Communication Scientifique Directe)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1294671590","host_organization_name":"Centre National de la Recherche Scientifique","host_organization_lineage":["https://openalex.org/I1294671590"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"IEEE Transactions on Information Theory, 2011, 57, pp.4026 - 4034. &#x27E8;10.1109/TIT.2011.2133710&#x27E9;","raw_type":"Journal articles"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":26,"referenced_works":["https://openalex.org/W254342579","https://openalex.org/W614581382","https://openalex.org/W1512650836","https://openalex.org/W1590757293","https://openalex.org/W1965457508","https://openalex.org/W1971749553","https://openalex.org/W1983874169","https://openalex.org/W1985813693","https://openalex.org/W1987805576","https://openalex.org/W1993155761","https://openalex.org/W1993562512","https://openalex.org/W1995875735","https://openalex.org/W2018127377","https://openalex.org/W2020374807","https://openalex.org/W2023711680","https://openalex.org/W2065947175","https://openalex.org/W2079108315","https://openalex.org/W2083682556","https://openalex.org/W2099111195","https://openalex.org/W2147758935","https://openalex.org/W2329077652","https://openalex.org/W2333466645","https://openalex.org/W3098134314","https://openalex.org/W4230736860","https://openalex.org/W4249000472","https://openalex.org/W4249446031"],"related_works":["https://openalex.org/W2037356780","https://openalex.org/W2045693826","https://openalex.org/W2898188192","https://openalex.org/W3114186922","https://openalex.org/W2300741437","https://openalex.org/W2159862004","https://openalex.org/W2398403988","https://openalex.org/W4224248810","https://openalex.org/W4205235378","https://openalex.org/W2330170324"],"abstract_inverted_index":{"We":[0],"study":[1,97],"entropy":[2,9,37,44,94,119,139,141],"rates":[3,95,120],"of":[4,43,69,135,163],"random":[5,70],"sequences":[6],"for":[7,35,39,143],"general":[8],"functionals":[10],"including":[11],"the":[12,19,27,36,49,61,93,127,133,136,164],"classical":[13],"Shannon":[14,116],"and":[15,18,23,65,140],"R\u00e9nyi":[16,118],"entropies":[17],"more":[20],"recent":[21],"Tsallis":[22],"Sharma\u2013Mittal":[24],"ones.":[25],"In":[26,126],"first":[28],"part,":[29,129],"we":[30,130],"obtain":[31],"an":[32],"explicit":[33,86],"formula":[34],"rate":[38,142],"a":[40,52,109,123,155,159],"large":[41],"class":[42],"functionals,":[45],"as":[46,48,60],"soon":[47],"process":[50],"satisfies":[51],"regularity":[53],"property":[54],"known":[55],"in":[56],"dynamical":[57],"systems":[58],"theory":[59],"quasi-power":[62],"property.":[63,75],"Independent":[64],"identically":[66],"distributed":[67],"sequence":[68],"variables":[71],"naturally":[72],"satisfy":[73,81],"this":[74],"Markov":[76,145],"chains":[77],"are":[78,98,113,152],"proven":[79],"to":[80,101,115,122],"it,":[82],"too,":[83],"under":[84,96],"simple":[85],"conditions":[87],"on":[88,132],"their":[89],"transition":[90],"probabilities.":[91],"All":[92],"thus":[99],"shown":[100],"be":[102],"either":[103],"infinite":[104],"or":[105,117],"zero":[106],"except":[107],"at":[108],"threshold":[110],"where":[111],"they":[112],"equal":[114],"up":[121],"multiplicative":[124],"constant.":[125],"second":[128],"focus":[131],"estimation":[134,162],"marginal":[137],"generalized":[138],"parametric":[144],"chains.":[146],"Estimators":[147],"with":[148],"good":[149],"asymptotic":[150],"properties":[151],"built":[153],"through":[154],"plug-in":[156],"procedure":[157],"using":[158],"maximum":[160],"likelihood":[161],"parameter.":[165]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":3},{"year":2023,"cited_by_count":2},{"year":2022,"cited_by_count":6},{"year":2021,"cited_by_count":4},{"year":2020,"cited_by_count":6},{"year":2019,"cited_by_count":1},{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":2},{"year":2016,"cited_by_count":1},{"year":2015,"cited_by_count":3},{"year":2012,"cited_by_count":1}],"updated_date":"2026-03-20T23:20:44.827607","created_date":"2025-10-10T00:00:00"}
