{"id":"https://openalex.org/W7220745724","doi":"https://doi.org/10.48550/arxiv.2610.05478","title":"From Counting to Continuous Natural Exponential Families","display_name":"From Counting to Continuous Natural Exponential Families","publication_year":2026,"publication_date":"2026-10-04","ids":{"openalex":"https://openalex.org/W7220745724","doi":"https://doi.org/10.48550/arxiv.2610.05478"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.05478","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.05478","pdf_url":"https://arxiv.org/pdf/2610.05478","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"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":null,"raw_type":"text"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2610.05478","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5068010330","display_name":"Shaul K. Bar\u2010Lev","orcid":"https://orcid.org/0000-0002-1770-7091","observed_orcids":["https://orcid.org/0000-0002-1770-7091"]},"institutions":[{"id":"https://openalex.org/I157943965","display_name":"Holon Institute of Technology","ror":"https://ror.org/02prqh017","country_code":"IL","type":"education","lineage":["https://openalex.org/I157943965"]}],"countries":["IL"],"is_corresponding":true,"raw_author_name":"Bar-Lev, Shaul K.","raw_affiliation_strings":["Faculty of Industrial Engineering and Technology Management HIT -Holon Institute of Technology , Holon , Israel"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Faculty of Industrial Engineering and Technology Management HIT -Holon Institute of Technology , Holon , Israel","institution_ids":["https://openalex.org/I157943965"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5068010330"],"corresponding_institution_ids":["https://openalex.org/I157943965"],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T13500","display_name":"Probability and Statistical Research","score":0.5845000147819519,"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"}},"topics":[{"id":"https://openalex.org/T13500","display_name":"Probability and Statistical Research","score":0.5845000147819519,"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/T10968","display_name":"Statistical Distribution Estimation and Applications","score":0.3564000129699707,"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/T11901","display_name":"Bayesian Methods and Mixture Models","score":0.017100000753998756,"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/natural-exponential-families","display_name":"natural exponential families","score":1.0},{"id":"https://openalex.org/keywords/variance-function","display_name":"variance function","score":0.9829999804496765},{"id":"https://openalex.org/keywords/infinitely-divisible-distributions","display_name":"infinitely divisible distributions","score":0.6880000233650208}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8736000061035156},{"id":"https://openalex.org/C118733216","wikidata":"https://www.wikidata.org/wiki/Q332504","display_name":"Absolute continuity","level":2,"score":0.6446999907493591},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.5400000214576721},{"id":"https://openalex.org/C2777105136","wikidata":"https://www.wikidata.org/wiki/Q827230","display_name":"Lebesgue measure","level":3,"score":0.5023000240325928},{"id":"https://openalex.org/C2780009758","wikidata":"https://www.wikidata.org/wiki/Q6804172","display_name":"Measure (data warehouse)","level":2,"score":0.47749999165534973},{"id":"https://openalex.org/C151376022","wikidata":"https://www.wikidata.org/wiki/Q168698","display_name":"Exponential function","level":2,"score":0.46619999408721924},{"id":"https://openalex.org/C2834757","wikidata":"https://www.wikidata.org/wiki/Q4925424","display_name":"Monotone polygon","level":2,"score":0.4560000002384186},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.45419999957084656},{"id":"https://openalex.org/C2780388253","wikidata":"https://www.wikidata.org/wiki/Q5421508","display_name":"Exponent","level":2,"score":0.4496999979019165},{"id":"https://openalex.org/C60775368","wikidata":"https://www.wikidata.org/wiki/Q17099337","display_name":"Natural exponential family","level":3,"score":0.42570000886917114},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.42179998755455017},{"id":"https://openalex.org/C55974624","wikidata":"https://www.wikidata.org/wiki/Q1188504","display_name":"Exponential family","level":2,"score":0.41909998655319214},{"id":"https://openalex.org/C36503486","wikidata":"https://www.wikidata.org/wiki/Q11235244","display_name":"Domain (mathematical analysis)","level":2,"score":0.4027999937534332},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3946000039577484},{"id":"https://openalex.org/C14158598","wikidata":"https://www.wikidata.org/wiki/Q853831","display_name":"Lebesgue integration","level":2,"score":0.3790999948978424},{"id":"https://openalex.org/C207467116","wikidata":"https://www.wikidata.org/wiki/Q4385666","display_name":"Inverse","level":2,"score":0.3707999885082245},{"id":"https://openalex.org/C120488996","wikidata":"https://www.wikidata.org/wiki/Q17016064","display_name":"Variance function","level":3,"score":0.35910001397132874},{"id":"https://openalex.org/C184264201","wikidata":"https://www.wikidata.org/wiki/Q21199","display_name":"Natural number","level":2,"score":0.3522000014781952},{"id":"https://openalex.org/C43994743","wikidata":"https://www.wikidata.org/wiki/Q516776","display_name":"Measurable function","level":3,"score":0.3370000123977661},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.32120001316070557},{"id":"https://openalex.org/C99844830","wikidata":"https://www.wikidata.org/wiki/Q102441924","display_name":"Scaling","level":2,"score":0.314300000667572},{"id":"https://openalex.org/C2780841128","wikidata":"https://www.wikidata.org/wiki/Q5073781","display_name":"Characterization (materials science)","level":2,"score":0.28679999709129333},{"id":"https://openalex.org/C143724316","wikidata":"https://www.wikidata.org/wiki/Q312468","display_name":"Series (stratigraphy)","level":2,"score":0.2865999937057495},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.2791999876499176},{"id":"https://openalex.org/C73905626","wikidata":"https://www.wikidata.org/wiki/Q206925","display_name":"Power series","level":2,"score":0.2770000100135803},{"id":"https://openalex.org/C122123141","wikidata":"https://www.wikidata.org/wiki/Q176623","display_name":"Random variable","level":2,"score":0.27399998903274536},{"id":"https://openalex.org/C2778199601","wikidata":"https://www.wikidata.org/wiki/Q5165476","display_name":"Continuous function (set theory)","level":3,"score":0.27399998903274536},{"id":"https://openalex.org/C16757182","wikidata":"https://www.wikidata.org/wiki/Q33456","display_name":"Power function","level":2,"score":0.27390000224113464},{"id":"https://openalex.org/C2778112365","wikidata":"https://www.wikidata.org/wiki/Q3511065","display_name":"Sequence (biology)","level":2,"score":0.2718000113964081},{"id":"https://openalex.org/C43323394","wikidata":"https://www.wikidata.org/wiki/Q3799278","display_name":"Lebesgue\u2013Stieltjes integration","level":5,"score":0.2639000117778778}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.05478","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.05478","pdf_url":"https://arxiv.org/pdf/2610.05478","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"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":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2610.05478","doi":"https://doi.org/10.48550/arxiv.2610.05478","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.05478","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2610.05478","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.05478","pdf_url":"https://arxiv.org/pdf/2610.05478","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"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":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7220745724.pdf","grobid_xml":"https://content.openalex.org/works/W7220745724.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W7046205214","https://openalex.org/W6910660516","https://openalex.org/W2024694330","https://openalex.org/W2757582478","https://openalex.org/W4411495014","https://openalex.org/W2194124705","https://openalex.org/W292386408","https://openalex.org/W2078907889","https://openalex.org/W2045692636","https://openalex.org/W2269482239"],"abstract_inverted_index":{"Natural":[0],"exponential":[1],"families":[2],"(NEFs)":[3],"are":[4],"characterized":[5],"by":[6,12],"their":[7],"variance":[8,20],"functions":[9],"(VFs),":[10],"namely":[11],"the":[13,19,25,30,37,46,56,90,108,125,138,143,155,183,187,197],"pair":[14],"\\((V,M)\\),":[15],"where":[16,168],"\\(V\\)":[17,45],"expresses":[18],"as":[21,66],"a":[22,42,69,98,129,171,181],"function":[23,44],"of":[24,48,55,68,92],"mean":[26,31,100,184],"and":[27,51,111,115,200],"\\(M\\)":[28],"is":[29,41,89,146,170,180,186],"domain.":[32,101],"This":[33,102],"characterization":[34],"naturally":[35],"raises":[36],"inverse":[38],"question:":[39],"when":[40,141],"given":[43],"VF":[47,91],"an":[49,93],"NEF,":[50],"what":[52],"structural":[53],"properties":[54],"family":[57],"can":[58],"be":[59],"read":[60],"directly":[61],"from":[62],"\\(V\\)?":[63],"Bar-Lev":[64],"(1987),":[65],"part":[67],"more":[70],"general":[71],"result":[72],"for":[73,196,201],"absolutely":[74,147],"monotone":[75],"functions,":[76],"showed":[77],"that":[78,123],"every":[79],"nonzero":[80,172],"polynomial":[81,103],"\\[":[82,165],"V(m)=\\sum_{j=1}^{r}":[83],"a_j":[84],"m^j,\\qquad":[85],"a_j\\ge":[86],"0,":[87],"\\]":[88,167],"infinitely":[94],"divisible":[95],"NEF":[96,131,145],"on":[97,133,154],"positive":[99,156,189],"class":[104],"splits":[105],"exhaustively":[106],"into":[107],"cases":[109],"\\(a_1>0\\)":[110],"\\(a_1=0\\).":[112],"Bar-Lev,":[113],"Letac":[114],"Ridder":[116],"(2024)":[117],"proved":[118],"that,":[119],"after":[120],"scaling":[121],"so":[122],"\\(a_1=1\\),":[124],"first":[126],"case":[127],"yields":[128],"counting":[130],"supported":[132],"\\(\\mathbb":[134],"N_0\\).":[135],"We":[136,191],"establish":[137],"complementary":[139],"result:":[140],"\\(a_1=0\\),":[142],"corresponding":[144],"continuous":[148],"with":[149,175],"respect":[150],"to":[151],"Lebesgue":[152],"measure":[153],"half-line.":[157,190],"More":[158],"generally,":[159],"we":[160],"prove":[161],"absolute":[162],"continuity":[163],"whenever":[164],"V(m)=m^2G(m),":[166],"\\(G\\)":[169,179],"power":[173],"series":[174],"nonnegative":[176],"coefficients.":[177],"When":[178],"polynomial,":[182],"domain":[185],"entire":[188],"also":[192],"record":[193],"several":[194],"consequences":[195],"cumulant":[198],"structure":[199],"statistical":[202],"inference.":[203]},"counts_by_year":[],"updated_date":"2026-11-06T12:05:46.655746","created_date":"2026-10-07T00:00:00"}