{"id":"https://openalex.org/W1896855243","doi":"https://doi.org/10.1109/tac.2011.2161834","title":"Kalman Filtering With Intermittent Observations: Weak Convergence to a Stationary Distribution","display_name":"Kalman Filtering With Intermittent Observations: Weak Convergence to a Stationary Distribution","publication_year":2011,"publication_date":"2011-07-20","ids":{"openalex":"https://openalex.org/W1896855243","doi":"https://doi.org/10.1109/tac.2011.2161834","mag":"1896855243"},"language":"en","primary_location":{"id":"doi:10.1109/tac.2011.2161834","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tac.2011.2161834","pdf_url":null,"source":{"id":"https://openalex.org/S184954342","display_name":"IEEE Transactions on Automatic Control","issn_l":"0018-9286","issn":["0018-9286","1558-2523","2334-3303"],"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 Automatic Control","raw_type":"journal-article"},"type":"preprint","indexed_in":["arxiv","crossref","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/0903.2890","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5077268766","display_name":"Soummya Kar","orcid":"https://orcid.org/0000-0002-8060-5581"},"institutions":[{"id":"https://openalex.org/I74973139","display_name":"Carnegie Mellon University","ror":"https://ror.org/05x2bcf33","country_code":"US","type":"education","lineage":["https://openalex.org/I74973139"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"S. Kar","raw_affiliation_strings":["Department of Electrical and Computer Engineering, Carnegie Mellon University, Pittsburgh, PA, USA","Dept. of Electr. & Comput. Eng., Carnegie Mellon Univ., Pittsburgh, PA, , USA"],"affiliations":[{"raw_affiliation_string":"Department of Electrical and Computer Engineering, Carnegie Mellon University, Pittsburgh, PA, USA","institution_ids":["https://openalex.org/I74973139"]},{"raw_affiliation_string":"Dept. of Electr. & Comput. Eng., Carnegie Mellon Univ., Pittsburgh, PA, , USA","institution_ids":["https://openalex.org/I74973139"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5006051783","display_name":"Bruno Sinopoli","orcid":"https://orcid.org/0000-0001-5778-4879"},"institutions":[{"id":"https://openalex.org/I74973139","display_name":"Carnegie Mellon University","ror":"https://ror.org/05x2bcf33","country_code":"US","type":"education","lineage":["https://openalex.org/I74973139"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"B. Sinopoli","raw_affiliation_strings":["Department of Electrical and Computer Engineering, Carnegie Mellon University, Pittsburgh, PA, USA","Dept. of Electr. & Comput. Eng., Carnegie Mellon Univ., Pittsburgh, PA, , USA"],"affiliations":[{"raw_affiliation_string":"Department of Electrical and Computer Engineering, Carnegie Mellon University, Pittsburgh, PA, USA","institution_ids":["https://openalex.org/I74973139"]},{"raw_affiliation_string":"Dept. of Electr. & Comput. Eng., Carnegie Mellon Univ., Pittsburgh, PA, , USA","institution_ids":["https://openalex.org/I74973139"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5045861415","display_name":"Jos\u00e9 M. F. Moura","orcid":"https://orcid.org/0000-0002-9822-8294"},"institutions":[{"id":"https://openalex.org/I74973139","display_name":"Carnegie Mellon University","ror":"https://ror.org/05x2bcf33","country_code":"US","type":"education","lineage":["https://openalex.org/I74973139"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"J. M. F. Moura","raw_affiliation_strings":["Department of Electrical and Computer Engineering, Carnegie Mellon University, Pittsburgh, PA, USA","Dept. of Electr. & Comput. Eng., Carnegie Mellon Univ., Pittsburgh, PA, , USA"],"affiliations":[{"raw_affiliation_string":"Department of Electrical and Computer Engineering, Carnegie Mellon University, Pittsburgh, PA, USA","institution_ids":["https://openalex.org/I74973139"]},{"raw_affiliation_string":"Dept. of Electr. & Comput. Eng., Carnegie Mellon Univ., Pittsburgh, PA, , USA","institution_ids":["https://openalex.org/I74973139"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5077268766"],"corresponding_institution_ids":["https://openalex.org/I74973139"],"apc_list":null,"apc_paid":null,"fwci":0.4284,"has_fulltext":true,"cited_by_count":7,"citation_normalized_percentile":{"value":0.70800878,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":95},"biblio":{"volume":"57","issue":"2","first_page":"405","last_page":"420"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10067","display_name":"Stochastic processes and financial applications","score":0.995199978351593,"subfield":{"id":"https://openalex.org/subfields/2003","display_name":"Finance"},"field":{"id":"https://openalex.org/fields/20","display_name":"Economics, Econometrics and Finance"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},"topics":[{"id":"https://openalex.org/T10067","display_name":"Stochastic processes and financial applications","score":0.995199978351593,"subfield":{"id":"https://openalex.org/subfields/2003","display_name":"Finance"},"field":{"id":"https://openalex.org/fields/20","display_name":"Economics, Econometrics and Finance"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},{"id":"https://openalex.org/T10711","display_name":"Target Tracking and Data Fusion in Sensor Networks","score":0.9836999773979187,"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/T11901","display_name":"Bayesian Methods and Mixture Models","score":0.9807000160217285,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.798763632774353},{"id":"https://openalex.org/keywords/sublinear-function","display_name":"Sublinear function","score":0.6387861967086792},{"id":"https://openalex.org/keywords/stationary-sequence","display_name":"Stationary sequence","score":0.566349983215332},{"id":"https://openalex.org/keywords/bernoulli-distribution","display_name":"Bernoulli distribution","score":0.5384592413902283},{"id":"https://openalex.org/keywords/ergodicity","display_name":"Ergodicity","score":0.5012459754943848},{"id":"https://openalex.org/keywords/weak-convergence","display_name":"Weak convergence","score":0.4820493161678314},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.47589266300201416},{"id":"https://openalex.org/keywords/bernoulli-process","display_name":"Bernoulli process","score":0.4708012342453003},{"id":"https://openalex.org/keywords/stationary-distribution","display_name":"Stationary distribution","score":0.4406850337982178},{"id":"https://openalex.org/keywords/convergence-of-random-variables","display_name":"Convergence of random variables","score":0.4398009181022644},{"id":"https://openalex.org/keywords/invariant","display_name":"Invariant (physics)","score":0.4382880926132202},{"id":"https://openalex.org/keywords/countable-set","display_name":"Countable set","score":0.43134626746177673},{"id":"https://openalex.org/keywords/stochastic-process","display_name":"Stochastic process","score":0.35959428548812866},{"id":"https://openalex.org/keywords/markov-chain","display_name":"Markov chain","score":0.3571286201477051},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.34519171714782715},{"id":"https://openalex.org/keywords/bernoullis-principle","display_name":"Bernoulli's principle","score":0.29750341176986694},{"id":"https://openalex.org/keywords/random-variable","display_name":"Random variable","score":0.21908792853355408},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.09336647391319275}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.798763632774353},{"id":"https://openalex.org/C117160843","wikidata":"https://www.wikidata.org/wiki/Q338652","display_name":"Sublinear function","level":2,"score":0.6387861967086792},{"id":"https://openalex.org/C18436578","wikidata":"https://www.wikidata.org/wiki/Q7604346","display_name":"Stationary sequence","level":3,"score":0.566349983215332},{"id":"https://openalex.org/C27956954","wikidata":"https://www.wikidata.org/wiki/Q391371","display_name":"Bernoulli distribution","level":3,"score":0.5384592413902283},{"id":"https://openalex.org/C201779956","wikidata":"https://www.wikidata.org/wiki/Q5426803","display_name":"Ergodicity","level":2,"score":0.5012459754943848},{"id":"https://openalex.org/C57945734","wikidata":"https://www.wikidata.org/wiki/Q7977941","display_name":"Weak convergence","level":3,"score":0.4820493161678314},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.47589266300201416},{"id":"https://openalex.org/C23990920","wikidata":"https://www.wikidata.org/wiki/Q518831","display_name":"Bernoulli process","level":3,"score":0.4708012342453003},{"id":"https://openalex.org/C98951983","wikidata":"https://www.wikidata.org/wiki/Q7604341","display_name":"Stationary distribution","level":3,"score":0.4406850337982178},{"id":"https://openalex.org/C95763700","wikidata":"https://www.wikidata.org/wiki/Q578985","display_name":"Convergence of random variables","level":3,"score":0.4398009181022644},{"id":"https://openalex.org/C190470478","wikidata":"https://www.wikidata.org/wiki/Q2370229","display_name":"Invariant (physics)","level":2,"score":0.4382880926132202},{"id":"https://openalex.org/C110729354","wikidata":"https://www.wikidata.org/wiki/Q185478","display_name":"Countable set","level":2,"score":0.43134626746177673},{"id":"https://openalex.org/C8272713","wikidata":"https://www.wikidata.org/wiki/Q176737","display_name":"Stochastic process","level":2,"score":0.35959428548812866},{"id":"https://openalex.org/C98763669","wikidata":"https://www.wikidata.org/wiki/Q176645","display_name":"Markov chain","level":2,"score":0.3571286201477051},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.34519171714782715},{"id":"https://openalex.org/C152361515","wikidata":"https://www.wikidata.org/wiki/Q181328","display_name":"Bernoulli's principle","level":2,"score":0.29750341176986694},{"id":"https://openalex.org/C122123141","wikidata":"https://www.wikidata.org/wiki/Q176623","display_name":"Random variable","level":2,"score":0.21908792853355408},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.09336647391319275},{"id":"https://openalex.org/C146978453","wikidata":"https://www.wikidata.org/wiki/Q3798668","display_name":"Aerospace engineering","level":1,"score":0.0},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.0},{"id":"https://openalex.org/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","level":0,"score":0.0},{"id":"https://openalex.org/C37914503","wikidata":"https://www.wikidata.org/wiki/Q156495","display_name":"Mathematical physics","level":1,"score":0.0},{"id":"https://openalex.org/C76178495","wikidata":"https://www.wikidata.org/wiki/Q4808784","display_name":"Asset (computer security)","level":2,"score":0.0},{"id":"https://openalex.org/C38652104","wikidata":"https://www.wikidata.org/wiki/Q3510521","display_name":"Computer security","level":1,"score":0.0}],"mesh":[],"locations_count":8,"locations":[{"id":"doi:10.1109/tac.2011.2161834","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tac.2011.2161834","pdf_url":null,"source":{"id":"https://openalex.org/S184954342","display_name":"IEEE Transactions on Automatic Control","issn_l":"0018-9286","issn":["0018-9286","1558-2523","2334-3303"],"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 Automatic Control","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:0903.2890","is_oa":true,"landing_page_url":"http://arxiv.org/abs/0903.2890","pdf_url":"https://arxiv.org/pdf/0903.2890","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"pmh:oai:figshare.com:article/6468947","is_oa":true,"landing_page_url":null,"pdf_url":"https://figshare.com/articles/Kalman_Filtering_with_Intermittent_Observations_Weak_Convergence_to_a_Stationary_Distribution/6468947","source":{"id":"https://openalex.org/S4377196282","display_name":"Figshare","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210132348","host_organization_name":"Figshare (United Kingdom)","host_organization_lineage":["https://openalex.org/I4210132348"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Text"},{"id":"mag:1896855243","is_oa":true,"landing_page_url":"https://arxiv.org/pdf/0903.2890.pdf","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"arXiv (Cornell University)","raw_type":null},{"id":"pmh:oai:repository.cmu.edu:ece-1333","is_oa":false,"landing_page_url":"http://repository.cmu.edu/ece/335","pdf_url":null,"source":{"id":"https://openalex.org/S4306400668","display_name":"Research Showcase @ Carnegie Mellon University (Carnegie Mellon University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I74973139","host_organization_name":"Carnegie Mellon University","host_organization_lineage":["https://openalex.org/I74973139"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Department of Electrical and Computer Engineering","raw_type":"text"},{"id":"doi:10.48550/arxiv.0903.2890","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.0903.2890","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article"},{"id":"doi:10.1184/r1/6468947","is_oa":true,"landing_page_url":"https://doi.org/10.1184/r1/6468947","pdf_url":null,"source":{"id":"https://openalex.org/S7407050927","display_name":"KiltHub Repository","issn_l":null,"issn":[],"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article-journal"},{"id":"doi:10.1184/r1/6468947.v1","is_oa":true,"landing_page_url":"https://doi.org/10.1184/r1/6468947.v1","pdf_url":null,"source":{"id":"https://openalex.org/S7407050927","display_name":"KiltHub Repository","issn_l":null,"issn":[],"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article-journal"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:0903.2890","is_oa":true,"landing_page_url":"http://arxiv.org/abs/0903.2890","pdf_url":"https://arxiv.org/pdf/0903.2890","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":37,"referenced_works":["https://openalex.org/W115967897","https://openalex.org/W261814290","https://openalex.org/W338436792","https://openalex.org/W361471363","https://openalex.org/W1986019586","https://openalex.org/W1988868967","https://openalex.org/W1994703549","https://openalex.org/W2035811299","https://openalex.org/W2064052856","https://openalex.org/W2065169455","https://openalex.org/W2071414465","https://openalex.org/W2081877251","https://openalex.org/W2095216716","https://openalex.org/W2102983250","https://openalex.org/W2105934661","https://openalex.org/W2113828476","https://openalex.org/W2118048279","https://openalex.org/W2135141230","https://openalex.org/W2137850460","https://openalex.org/W2142044905","https://openalex.org/W2152898444","https://openalex.org/W2156216606","https://openalex.org/W2156701190","https://openalex.org/W2159591388","https://openalex.org/W2162850525","https://openalex.org/W2164522996","https://openalex.org/W2166968389","https://openalex.org/W2172180431","https://openalex.org/W2482944896","https://openalex.org/W2745371750","https://openalex.org/W2963435958","https://openalex.org/W3140352127","https://openalex.org/W4210379104","https://openalex.org/W4292027056","https://openalex.org/W4293282250","https://openalex.org/W4297584681","https://openalex.org/W6680732042"],"related_works":["https://openalex.org/W2091896264","https://openalex.org/W2156668047","https://openalex.org/W596903966","https://openalex.org/W1994165559","https://openalex.org/W2963726397","https://openalex.org/W289853301","https://openalex.org/W2002513882","https://openalex.org/W3038039500","https://openalex.org/W2082531674","https://openalex.org/W2338668558","https://openalex.org/W115397153","https://openalex.org/W2552575716","https://openalex.org/W2014871121","https://openalex.org/W2091012202","https://openalex.org/W2951453895","https://openalex.org/W2611427719","https://openalex.org/W2090704469","https://openalex.org/W288678087","https://openalex.org/W2954854410","https://openalex.org/W610704983"],"abstract_inverted_index":{"The":[0],"paper":[1],"studies":[2],"the":[3,18,21,36,66,70,72,125,146,150,154,158,161,166,182,190,194,197,201,207,215,223,230],"asymptotic":[4,67],"behavior":[5],"of":[6,20,69,74,149,160,169,173,184,193,196,206,214,229],"discrete":[7],"time":[8],"Random":[9],"Riccati":[10],"Equations":[11],"(RRE)":[12],"arising":[13],"in":[14,112,135,176,181],"Kalman":[15],"filtering":[16],"when":[17],"arrival":[19,109,121],"observations":[22],"is":[23,178,225],"described":[24],"by":[25],"a":[26,48,55,83,141,170,226],"Bernoulli":[27],"independent":[28],"and":[29,53,94,200],"identically":[30],"distributed":[31],"(i.i.d.)":[32],"process.":[33],"We":[34,144,188],"model":[35],"RRE":[37,155],"as":[38,165],"an":[39],"order-preserving,":[40,60],"strongly":[41,61],"sublinear":[42,62],"random":[43,75],"dynamical":[44],"system":[45],"(RDS).":[46],"Under":[47],"sufficient":[49],"condition,":[50],"stochastic":[51,97],"boundedness,":[52],"using":[54],"limit-set":[56],"dichotomy":[57],"result":[58],"for":[59,104,128],"RDS,":[63],"we":[64,114],"establish":[65,116],"properties":[68],"RRE:":[71],"sequence":[73],"prediction":[76],"error":[77],"covariance":[78],"matrices":[79],"converges":[80],"weakly":[81],"to":[82,156,210],"unique":[84],"invariant":[85,133,163,198,216],"distribution,":[86],"whose":[87],"support":[88,159,195,224],"exhibits":[89],"fractal":[90],"behavior.":[91],"For":[92],"stabilizable":[93],"detectable":[95],"systems,":[96],"boundedness":[98],"(and":[99],"hence":[100],"weak":[101,117],"convergence)":[102],"holds":[103],"any":[105],"nonzero":[106],"observation":[107],"packet":[108],"probability":[110,127],"and,":[111],"particular,":[113],"can":[115],"convergence":[118],"at":[119],"operating":[120],"rates":[122],"well":[123],"below":[124],"critical":[126],"mean":[129],"stability":[130],"(the":[131],"resulting":[132],"measure":[134],"that":[136,222],"situation":[137],"does":[138],"not":[139,179],"possess":[140],"first":[142],"moment).":[143],"apply":[145],"weak-Feller":[147],"property":[148],"Markov":[151],"process":[152],"governing":[153],"characterize":[157],"limiting":[162],"distribution":[164,199],"topological":[167],"closure":[168],"countable":[171],"set":[172,183],"points,":[174],"which,":[175],"general,":[177],"dense":[180],"positive":[185],"semi-definite":[186],"matrices.":[187],"use":[189],"explicit":[191],"characterization":[192],"almost":[202],"sure":[203],"(a.s.)":[204],"ergodicity":[205],"sample":[208],"paths":[209],"easily":[211],"compute":[212],"statistics":[213],"distribution.":[217],"A":[218],"one-dimensional":[219],"example":[220],"illustrates":[221],"fractured":[227],"subset":[228],"non-negative":[231],"reals":[232],"with":[233],"self-similarity":[234],"properties.":[235]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2018,"cited_by_count":1}],"updated_date":"2026-03-20T23:20:44.827607","created_date":"2025-10-10T00:00:00"}