{"id":"https://openalex.org/W2992677118","doi":"https://doi.org/10.3390/sym11121487","title":"A Note on Distributions in the Second Chaos","display_name":"A Note on Distributions in the Second Chaos","publication_year":2019,"publication_date":"2019-12-06","ids":{"openalex":"https://openalex.org/W2992677118","doi":"https://doi.org/10.3390/sym11121487","mag":"2992677118"},"language":"en","primary_location":{"id":"doi:10.3390/sym11121487","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym11121487","pdf_url":"https://www.mdpi.com/2073-8994/11/12/1487/pdf?version=1576298067","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2073-8994/11/12/1487/pdf?version=1576298067","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5021402082","display_name":"Pauliina Ilmonen","orcid":"https://orcid.org/0000-0002-8992-0553"},"institutions":[{"id":"https://openalex.org/I9927081","display_name":"Aalto University","ror":"https://ror.org/020hwjq30","country_code":"FI","type":"education","lineage":["https://openalex.org/I9927081"]}],"countries":["FI"],"is_corresponding":false,"raw_author_name":"Pauliina Ilmonen","raw_affiliation_strings":["Department of Mathematics and Systems Analysis, Aalto University School of Science, 00076 Aalto, Finland"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Systems Analysis, Aalto University School of Science, 00076 Aalto, Finland","institution_ids":["https://openalex.org/I9927081"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5027821152","display_name":"Lauri Viitasaari","orcid":"https://orcid.org/0000-0002-3015-8664"},"institutions":[{"id":"https://openalex.org/I9927081","display_name":"Aalto University","ror":"https://ror.org/020hwjq30","country_code":"FI","type":"education","lineage":["https://openalex.org/I9927081"]}],"countries":["FI"],"is_corresponding":true,"raw_author_name":"Lauri Viitasaari","raw_affiliation_strings":["Department of Information and Service Management, Aalto University School of Business, 00076 Aalto, Finland"],"raw_orcid":"https://orcid.org/0000-0002-3015-8664","affiliations":[{"raw_affiliation_string":"Department of Information and Service Management, Aalto University School of Business, 00076 Aalto, Finland","institution_ids":["https://openalex.org/I9927081"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5027821152"],"corresponding_institution_ids":["https://openalex.org/I9927081"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.23788223,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"11","issue":"12","first_page":"1487","last_page":"1487"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10067","display_name":"Stochastic processes and financial applications","score":0.9980999827384949,"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":0.9980999827384949,"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/T11152","display_name":"Stochastic processes and statistical mechanics","score":0.9915000200271606,"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/T10282","display_name":"Financial Risk and Volatility Modeling","score":0.9894000291824341,"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/smoothness","display_name":"Smoothness","score":0.8395475149154663},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6514458060264587},{"id":"https://openalex.org/keywords/representation","display_name":"Representation (politics)","score":0.6186672449111938},{"id":"https://openalex.org/keywords/chaos","display_name":"CHAOS (operating system)","score":0.6170784831047058},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.5107012391090393},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.49689677357673645},{"id":"https://openalex.org/keywords/meaning","display_name":"Meaning (existential)","score":0.445563942193985},{"id":"https://openalex.org/keywords/probability-density-function","display_name":"Probability density function","score":0.4294623136520386},{"id":"https://openalex.org/keywords/random-variable","display_name":"Random variable","score":0.41506949067115784},{"id":"https://openalex.org/keywords/statistical-physics","display_name":"Statistical physics","score":0.4119046628475189},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.32627689838409424},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.2978132665157318},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.20568469166755676},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.14855855703353882},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.14816799759864807},{"id":"https://openalex.org/keywords/philosophy","display_name":"Philosophy","score":0.05816003680229187}],"concepts":[{"id":"https://openalex.org/C102634674","wikidata":"https://www.wikidata.org/wiki/Q868473","display_name":"Smoothness","level":2,"score":0.8395475149154663},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6514458060264587},{"id":"https://openalex.org/C2776359362","wikidata":"https://www.wikidata.org/wiki/Q2145286","display_name":"Representation (politics)","level":3,"score":0.6186672449111938},{"id":"https://openalex.org/C2779374083","wikidata":"https://www.wikidata.org/wiki/Q5011038","display_name":"CHAOS (operating system)","level":2,"score":0.6170784831047058},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.5107012391090393},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.49689677357673645},{"id":"https://openalex.org/C2780876879","wikidata":"https://www.wikidata.org/wiki/Q3054749","display_name":"Meaning (existential)","level":2,"score":0.445563942193985},{"id":"https://openalex.org/C197055811","wikidata":"https://www.wikidata.org/wiki/Q207522","display_name":"Probability density function","level":2,"score":0.4294623136520386},{"id":"https://openalex.org/C122123141","wikidata":"https://www.wikidata.org/wiki/Q176623","display_name":"Random variable","level":2,"score":0.41506949067115784},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.4119046628475189},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.32627689838409424},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.2978132665157318},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.20568469166755676},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.14855855703353882},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.14816799759864807},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.05816003680229187},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C17744445","wikidata":"https://www.wikidata.org/wiki/Q36442","display_name":"Political science","level":0,"score":0.0},{"id":"https://openalex.org/C94625758","wikidata":"https://www.wikidata.org/wiki/Q7163","display_name":"Politics","level":2,"score":0.0},{"id":"https://openalex.org/C199539241","wikidata":"https://www.wikidata.org/wiki/Q7748","display_name":"Law","level":1,"score":0.0},{"id":"https://openalex.org/C78458016","wikidata":"https://www.wikidata.org/wiki/Q840400","display_name":"Evolutionary biology","level":1,"score":0.0},{"id":"https://openalex.org/C111472728","wikidata":"https://www.wikidata.org/wiki/Q9471","display_name":"Epistemology","level":1,"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":2,"locations":[{"id":"doi:10.3390/sym11121487","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym11121487","pdf_url":"https://www.mdpi.com/2073-8994/11/12/1487/pdf?version=1576298067","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},{"id":"pmh:oai:aaltodoc.aalto.fi:123456789/41961","is_oa":true,"landing_page_url":"https://research.aalto.fi/en/publications/2d8bcac1-89f2-42a8-8890-efc92f4c2bd6","pdf_url":null,"source":{"id":"https://openalex.org/S4306401663","display_name":"Aaltodoc (Aalto University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I9927081","host_organization_name":"Aalto University","host_organization_lineage":["https://openalex.org/I9927081"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"publishedVersion"}],"best_oa_location":{"id":"doi:10.3390/sym11121487","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym11121487","pdf_url":"https://www.mdpi.com/2073-8994/11/12/1487/pdf?version=1576298067","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2992677118.pdf","grobid_xml":"https://content.openalex.org/works/W2992677118.grobid-xml"},"referenced_works_count":24,"referenced_works":["https://openalex.org/W48692188","https://openalex.org/W113145852","https://openalex.org/W387259284","https://openalex.org/W629086490","https://openalex.org/W1483560004","https://openalex.org/W1508427692","https://openalex.org/W1591798773","https://openalex.org/W1711937294","https://openalex.org/W2028272962","https://openalex.org/W2031390992","https://openalex.org/W2068592427","https://openalex.org/W2099176558","https://openalex.org/W2100676242","https://openalex.org/W2156345480","https://openalex.org/W2481769506","https://openalex.org/W2498994366","https://openalex.org/W2591972302","https://openalex.org/W2963811485","https://openalex.org/W3010098330","https://openalex.org/W3099344973","https://openalex.org/W3124809427","https://openalex.org/W6604675954","https://openalex.org/W6657608170","https://openalex.org/W6785851439"],"related_works":["https://openalex.org/W2355760814","https://openalex.org/W2028225975","https://openalex.org/W2373670823","https://openalex.org/W2391213251","https://openalex.org/W2076078510","https://openalex.org/W2365843888","https://openalex.org/W2242259016","https://openalex.org/W2364639542","https://openalex.org/W2103565019","https://openalex.org/W2363158203"],"abstract_inverted_index":{"In":[0,70],"this":[1],"article":[2],"we":[3,59,72],"study":[4,63],"basic":[5],"properties":[6],"of":[7,66,76],"random":[8],"variables":[9],"X,":[10],"and":[11],"their":[12],"associated":[13],"distributions,":[14],"in":[15,98],"the":[16,55,64,74,99],"second":[17],"chaos,":[18],"meaning":[19],"that":[20],"X":[21,25],"has":[22],"a":[23,77],"representation":[24],"=":[26],"\u2211":[27],"k":[28,32,35,43,86],"\u2265":[29],"1":[30,38,49],"\u03bb":[31,85],"(":[33,46],"\u03be":[34,42],"2":[36],"\u2212":[37],")":[39,50],",":[40,48],"where":[41],"\u223c":[44],"N":[45],"0":[47,88],"are":[51],"independent.":[52],"We":[53],"compute":[54],"L\u00e9vy-Khintchine":[56],"representations":[57],"which":[58],"then":[60],"use":[61],"to":[62],"smoothness":[65],"each":[67],"density":[68,79],"function.":[69],"particular,":[71],"prove":[73],"existence":[75],"smooth":[78],"with":[80],"asymptotically":[81],"vanishing":[82],"derivatives":[83],"whenever":[84],"\u2260":[87],"infinitely":[89],"often.":[90],"Our":[91],"work":[92],"generalises":[93],"some":[94],"known":[95],"results":[96],"presented":[97],"literature.":[100]},"counts_by_year":[],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}