{"id":"https://openalex.org/W2110971055","doi":"https://doi.org/10.1109/tit.1969.1054387","title":"Characteristics of Gaussian random processes by representations in terms of independent random variables","display_name":"Characteristics of Gaussian random processes by representations in terms of independent random variables","publication_year":1969,"publication_date":"1969-11-01","ids":{"openalex":"https://openalex.org/W2110971055","doi":"https://doi.org/10.1109/tit.1969.1054387","mag":"2110971055"},"language":"en","primary_location":{"id":"doi:10.1109/tit.1969.1054387","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tit.1969.1054387","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":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5021505026","display_name":"Percy A. Pierre","orcid":null},"institutions":[{"id":"https://openalex.org/I1309849503","display_name":"RAND Corporation","ror":"https://ror.org/00f2z7n96","country_code":"US","type":"nonprofit","lineage":["https://openalex.org/I1309849503"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"P. Pierre","raw_affiliation_strings":["System Sciences Department, RAND Corporation, Santa Monica, CA, USA"],"affiliations":[{"raw_affiliation_string":"System Sciences Department, RAND Corporation, Santa Monica, CA, USA","institution_ids":["https://openalex.org/I1309849503"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5021505026"],"corresponding_institution_ids":["https://openalex.org/I1309849503"],"apc_list":null,"apc_paid":null,"fwci":14.2543,"has_fulltext":false,"cited_by_count":12,"citation_normalized_percentile":{"value":0.98764585,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":94,"max":96},"biblio":{"volume":"15","issue":"6","first_page":"648","last_page":"658"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T13244","display_name":"Analysis of environmental and stochastic processes","score":0.9962000250816345,"subfield":{"id":"https://openalex.org/subfields/2308","display_name":"Management, Monitoring, Policy and Law"},"field":{"id":"https://openalex.org/fields/23","display_name":"Environmental Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T13244","display_name":"Analysis of environmental and stochastic processes","score":0.9962000250816345,"subfield":{"id":"https://openalex.org/subfields/2308","display_name":"Management, Monitoring, Policy and Law"},"field":{"id":"https://openalex.org/fields/23","display_name":"Environmental Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11871","display_name":"Advanced Statistical Methods and Models","score":0.9879999756813049,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T13487","display_name":"Statistical and numerical algorithms","score":0.9563999772071838,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/isospectral","display_name":"Isospectral","score":0.5786834955215454},{"id":"https://openalex.org/keywords/random-variable","display_name":"Random variable","score":0.47903385758399963},{"id":"https://openalex.org/keywords/gaussian","display_name":"Gaussian","score":0.4761657416820526},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.39538493752479553},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.38949620723724365},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.3771631121635437},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.3381633758544922},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.3320434093475342},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.22208648920059204},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.21911370754241943},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.1512986123561859}],"concepts":[{"id":"https://openalex.org/C2779710842","wikidata":"https://www.wikidata.org/wiki/Q6086323","display_name":"Isospectral","level":2,"score":0.5786834955215454},{"id":"https://openalex.org/C122123141","wikidata":"https://www.wikidata.org/wiki/Q176623","display_name":"Random variable","level":2,"score":0.47903385758399963},{"id":"https://openalex.org/C163716315","wikidata":"https://www.wikidata.org/wiki/Q901177","display_name":"Gaussian","level":2,"score":0.4761657416820526},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.39538493752479553},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.38949620723724365},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.3771631121635437},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.3381633758544922},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.3320434093475342},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.22208648920059204},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.21911370754241943},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.1512986123561859},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1109/tit.1969.1054387","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tit.1969.1054387","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"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":14,"referenced_works":["https://openalex.org/W208843157","https://openalex.org/W1964910808","https://openalex.org/W2007641452","https://openalex.org/W2045463928","https://openalex.org/W2049917002","https://openalex.org/W2060900941","https://openalex.org/W2078131860","https://openalex.org/W2084497560","https://openalex.org/W2120191353","https://openalex.org/W2325351015","https://openalex.org/W2477688680","https://openalex.org/W2798125939","https://openalex.org/W2798461040","https://openalex.org/W2799137445"],"related_works":["https://openalex.org/W2327570786","https://openalex.org/W4327845731","https://openalex.org/W2951065424","https://openalex.org/W1563360693","https://openalex.org/W4225492786","https://openalex.org/W2950476590","https://openalex.org/W39207448","https://openalex.org/W1991776635","https://openalex.org/W1979823253","https://openalex.org/W1980475149"],"abstract_inverted_index":{"Certain":[0],"isospectral":[1,67],"classes":[2,64,68],"of":[3,10,22,33,40,112,115,137,142,151,167,171,197,201,213,216],"random":[4,199,202],"processes":[5,12,144,193],"and":[6,207],"certain":[7,191],"linear":[8,38,214],"representations":[9],"these":[11],"are":[13,69,107,204,219],"considered.":[14,205],"It":[15,97,159],"is":[16,25,98,131,146],"shown":[17],"that":[18,100,145],"the":[19,26,30,54,89,102,152,165,168,211],"Gaussian":[20,55,143,153],"member":[21],"each":[23,111],"class":[24],"only":[27,53],"one":[28],"having":[29],"desirable":[31],"property":[32],"being":[34],"representable":[35],"as":[36,183,195],"a":[37,77,113,138,149,162,172,198],"combination":[39],"statistically":[41],"independent":[42,59,109],"components.":[43],"For":[44,72],"example,":[45,73],"among":[46,63,155],"all":[47],"strictly":[48],"stationary,":[49],"band-limited,":[50],"white-noise":[51],"processes,":[52],"process":[56,218],"has":[57],"mutually":[58,108],"Nyquist":[60],"samples.":[61],"Characterizations":[62],"wider":[65],"than":[66],"also":[70,160],"obtained.":[71,220],"let":[74],"x(t)":[75],"have":[76],"rational":[78],"spectral":[79],"density":[80],"with":[81,120],"Karhunen-Lo\u00e8ve":[82],"(K-L)":[83],"coefficients":[84,103],"<tex":[85,91,104,117,121,128,175,184],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[86,92,105,118,122,129,176,185],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">\\{a_{i}(T_{k})\\}</tex>":[87,106],"for":[88,110,210],"interval":[90],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">[-":[93],"T_{k},":[94],"I_{k}]</tex>":[95],".":[96,189],"argued":[99],"if":[101],"sequence":[114],"intervals":[116],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">\\{T_{k}\\}</tex>":[119],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">T_{k}":[123,186],"\\rightarrow":[124,187],"\\infty</tex>":[125,188],",":[126],"then":[127],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">x(t)</tex>":[130],"Gaussian.":[132],"This":[133],"conclusion":[134],"makes":[135],"use":[136],"more":[139],"general":[140],"characterization":[141,150],"obtained":[147],"using":[148],"distribution":[154],"infinitely":[156],"divisible":[157],"distributions.":[158],"uses":[161],"conjecture":[163],"about":[164],"behavior":[166],"K-L":[169],"representation":[170],"known":[173],"function":[174],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">m(t),":[177],"t":[178],"\\in":[179],"(-":[180],"\\infty,":[181],"\\infty)</tex>":[182],"Finally,":[190],"non-Gaussian":[192],"defined":[194],"sums":[196],"number":[200],"pulses":[203],"Necessary":[206],"sufficient":[208],"conditions":[209],"independence":[212],"functionals":[215],"this":[217]},"counts_by_year":[{"year":2024,"cited_by_count":2},{"year":2023,"cited_by_count":2}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}