{"id":"https://openalex.org/W1985358988","doi":"https://doi.org/10.1214/aop/1176997030","title":"Sample Functions of the $N$-Parameter Wiener Process","display_name":"Sample Functions of the $N$-Parameter Wiener Process","publication_year":1973,"publication_date":"1973-02-01","ids":{"openalex":"https://openalex.org/W1985358988","doi":"https://doi.org/10.1214/aop/1176997030","mag":"1985358988"},"language":"en","primary_location":{"id":"doi:10.1214/aop/1176997030","doi":"https://doi.org/10.1214/aop/1176997030","is_oa":true,"landing_page_url":"https://doi.org/10.1214/aop/1176997030","pdf_url":"https://projecteuclid.org/journals/annals-of-probability/volume-1/issue-1/Sample-Functions-of-the-N-Parameter-Wiener-Process/10.1214/aop/1176997030.pdf","source":{"id":"https://openalex.org/S40212770","display_name":"The Annals of Probability","issn_l":"0091-1798","issn":["0091-1798","2168-894X"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["abdc-a-star","anvur-class-a","anvur-scientific","cwts-core","dongbi-a","jufo-3","norway-2","poland-140","russia-white-list-1"],"host_organization":"https://openalex.org/P4310319881","host_organization_name":"Institute of Mathematical Statistics","host_organization_lineage":["https://openalex.org/P4310319881"],"host_organization_lineage_names":["Institute of Mathematical Statistics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Annals of Probability","raw_type":"journal-article"},"volume":null,"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"bronze","oa_url":"https://projecteuclid.org/journals/annals-of-probability/volume-1/issue-1/Sample-Functions-of-the-N-Parameter-Wiener-Process/10.1214/aop/1176997030.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5005623147","display_name":"Steven Orey","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Steven Orey","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5047207299","display_name":"William E. Pruitt","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"William E. Pruitt","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":3.633,"has_fulltext":false,"cited_by_count":154,"citation_normalized_percentile":{"value":0.94374643,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":88,"max":98},"biblio":{"volume":"1","issue":"1","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11152","display_name":"Stochastic processes and statistical mechanics","score":0.7163000106811523,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11152","display_name":"Stochastic processes and statistical mechanics","score":0.7163000106811523,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"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/T13500","display_name":"Probability and Statistical Research","score":0.20640000700950623,"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/T10067","display_name":"Stochastic processes and financial applications","score":0.016499999910593033,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/recurrence-properties","display_name":"recurrence properties","score":0.6850000023841858},{"id":"https://openalex.org/keywords/integral-test","display_name":"integral test","score":0.7020000219345093},{"id":"https://openalex.org/keywords/gaussian-processes","display_name":"Gaussian processes","score":0.6340000033378601}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.9075020551681519},{"id":"https://openalex.org/C60391097","wikidata":"https://www.wikidata.org/wiki/Q1056809","display_name":"Wiener process","level":2,"score":0.6406846046447754},{"id":"https://openalex.org/C61326573","wikidata":"https://www.wikidata.org/wiki/Q1496376","display_name":"Gaussian process","level":3,"score":0.5019664764404297},{"id":"https://openalex.org/C137250428","wikidata":"https://www.wikidata.org/wiki/Q5178897","display_name":"Covariance function","level":3,"score":0.4950956404209137},{"id":"https://openalex.org/C178650346","wikidata":"https://www.wikidata.org/wiki/Q201984","display_name":"Covariance","level":2,"score":0.46901434659957886},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4672538936138153},{"id":"https://openalex.org/C163716315","wikidata":"https://www.wikidata.org/wiki/Q901177","display_name":"Gaussian","level":2,"score":0.4170233905315399},{"id":"https://openalex.org/C2777686260","wikidata":"https://www.wikidata.org/wiki/Q144037","display_name":"Calculus (dental)","level":2,"score":0.38423681259155273},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.38198158144950867},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.213498055934906},{"id":"https://openalex.org/C71924100","wikidata":"https://www.wikidata.org/wiki/Q11190","display_name":"Medicine","level":0,"score":0.0},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C199343813","wikidata":"https://www.wikidata.org/wiki/Q12128","display_name":"Dentistry","level":1,"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:10.1214/aop/1176997030","doi":"https://doi.org/10.1214/aop/1176997030","is_oa":true,"landing_page_url":"https://doi.org/10.1214/aop/1176997030","pdf_url":"https://projecteuclid.org/journals/annals-of-probability/volume-1/issue-1/Sample-Functions-of-the-N-Parameter-Wiener-Process/10.1214/aop/1176997030.pdf","source":{"id":"https://openalex.org/S40212770","display_name":"The Annals of Probability","issn_l":"0091-1798","issn":["0091-1798","2168-894X"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["abdc-a-star","anvur-class-a","anvur-scientific","cwts-core","dongbi-a","jufo-3","norway-2","poland-140","russia-white-list-1"],"host_organization":"https://openalex.org/P4310319881","host_organization_name":"Institute of Mathematical Statistics","host_organization_lineage":["https://openalex.org/P4310319881"],"host_organization_lineage_names":["Institute of Mathematical Statistics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Annals of Probability","raw_type":"journal-article"},{"id":"pmh:oai:CULeuclid:euclid.aop/1176997030","doi":null,"is_oa":false,"landing_page_url":"http://projecteuclid.org/euclid.aop/1176997030","pdf_url":null,"source":{"id":"https://openalex.org/S4306400787","display_name":"Project Euclid (Cornell University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":"","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.1214/aop/1176997030","doi":"https://doi.org/10.1214/aop/1176997030","is_oa":true,"landing_page_url":"https://doi.org/10.1214/aop/1176997030","pdf_url":"https://projecteuclid.org/journals/annals-of-probability/volume-1/issue-1/Sample-Functions-of-the-N-Parameter-Wiener-Process/10.1214/aop/1176997030.pdf","source":{"id":"https://openalex.org/S40212770","display_name":"The Annals of Probability","issn_l":"0091-1798","issn":["0091-1798","2168-894X"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["abdc-a-star","anvur-class-a","anvur-scientific","cwts-core","dongbi-a","jufo-3","norway-2","poland-140","russia-white-list-1"],"host_organization":"https://openalex.org/P4310319881","host_organization_name":"Institute of Mathematical Statistics","host_organization_lineage":["https://openalex.org/P4310319881"],"host_organization_lineage_names":["Institute of Mathematical Statistics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Annals of Probability","raw_type":"journal-article"},"sustainable_development_goals":[],"sustainable_development_goals_aurora":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W1985358988.pdf","grobid_xml":"https://content.openalex.org/works/W1985358988.grobid-xml"},"referenced_works_count":12,"referenced_works":["https://openalex.org/W1544367389","https://openalex.org/W1992921902","https://openalex.org/W1997465361","https://openalex.org/W2000470783","https://openalex.org/W2021851424","https://openalex.org/W2028737966","https://openalex.org/W2029078670","https://openalex.org/W2039264593","https://openalex.org/W2068286590","https://openalex.org/W2075531533","https://openalex.org/W2090647925","https://openalex.org/W2094081378"],"related_works":["https://openalex.org/W2154788797","https://openalex.org/W3002473118","https://openalex.org/W2951088868","https://openalex.org/W2375962503","https://openalex.org/W4294029521","https://openalex.org/W4321602985","https://openalex.org/W2063356007","https://openalex.org/W2072847798","https://openalex.org/W2949256156","https://openalex.org/W4299499750"],"abstract_inverted_index":{"Let":[0],"$W^{(N)}$":[1],"denote":[2],"the":[3,48,63,76,91,99,114],"$N$-parameter":[4,60],"Wiener":[5,61],"process,":[6,62],"that":[7,116],"is":[8,45,69,118],"a":[9],"real-valued":[10],"Gaussian":[11],"process":[12,49],"with":[13,50,70],"zero":[14],"means":[15],"and":[16,72,104,109,124],"covariance":[17],"$\\Pi^N_{i=1}":[18],"(s_i":[19],"\\wedge":[20],"t_i)$":[21],"where":[22],"$s":[23],"=":[24,28,38,94,96],"\\langle":[25,29],"s_i\\rangle,":[26],"t":[27],"t_i\\rangle,":[30],"s_i":[31],"\\geqq":[32,35],"0,":[33,36],"t_i":[34],"i":[37],"1,":[39],"2,":[40],"\\cdots":[41],"N$.":[42],"Then":[43],"$W^{(N,d)}$":[44,117],"to":[46,98],"be":[47],"values":[51],"in":[52,90],"$R^d$":[53],"determined":[54],"by":[55],"making":[56],"each":[57],"component":[58],"an":[59],"components":[64],"being":[65],"independent.":[66],"Our":[67],"concern":[68],"continuity":[71],"recurrence":[73],"properties":[74],"of":[75,102,105,113],"sample":[77],"functions.":[78],"In":[79],"particular":[80],"we":[81],"give":[82],"integral":[83,100],"tests":[84,101],"for":[85],"upper":[86],"functions":[87],"which":[88],"reduce":[89],"case":[92],"$N":[93],"d":[95],"1$":[97],"Kolmogorov,":[103],"Chung-Erdos-Sirao.":[106],"We":[107],"formulate":[108],"prove":[110],"precise":[111],"statements":[112],"fact":[115],"interval":[119],"recurrent":[120],"(point":[121],"recurrent)":[122],"if":[123,126],"only":[125],"$d":[127],"\\leqq":[128],"2N":[129],"(d":[130],"<":[131],"2N)$.":[132]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":3},{"year":2022,"cited_by_count":2},{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":2},{"year":2019,"cited_by_count":5},{"year":2017,"cited_by_count":2},{"year":2016,"cited_by_count":1},{"year":2015,"cited_by_count":1},{"year":2014,"cited_by_count":4},{"year":2013,"cited_by_count":4},{"year":2012,"cited_by_count":4}],"updated_date":"2026-10-12T09:40:04.860316","created_date":"2025-10-10T00:00:00"}