{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,20]],"date-time":"2025-09-20T21:57:32Z","timestamp":1758405452640},"reference-count":12,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2008,10,24]],"date-time":"2008-10-24T00:00:00Z","timestamp":1224806400000},"content-version":"unspecified","delay-in-days":11376,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Proc. Camb. Phil. Soc."],"published-print":{"date-parts":[[1977,9]]},"abstract":"<jats:p>Goldie (2), Steutel (8, 9), Kelker (4), Keilson and Steutel (3) and several others have studied the mixtures of certain distributions which are infinitely divisible. Recently Shanbhag and Sreehari (7) have proved that if <jats:italic>Z<\/jats:italic> is exponential with unit parameter and for 0 &lt; \u03b1 &lt; 1, if <jats:italic>Y<jats:sub>x<\/jats:sub><\/jats:italic> is a positive stable random variable with <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0305004100053925_inline1\" \/>, <jats:italic>t<\/jats:italic> \u2265 0 and independent of <jats:italic>Z<\/jats:italic>, then for every 0 &lt; \u03b1 &lt; 1<\/jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0305004100053925_eqn1\" \/><\/jats:disp-formula><\/jats:p><jats:p>Using this result, they have obtained several interesting results concerning stable random variables including some extensions of the results of the above authors. More recently, Williams (11) has used the same approach to show that if <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0305004100053925_inline2\" \/>, where <jats:italic>n<\/jats:italic> is a positive integer \u2265 2, then <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0305004100053925_inline3\" \/> is distributed as the product of <jats:italic>n<\/jats:italic> \u2212 1 independent gamma random variables with index parameters \u03b1, 2\u03b1, \u2026, (<jats:italic>n<\/jats:italic> \u2212 1) \u03b1. Prior to these investigations, Zolotarev (12) had studied the problems of <jats:italic>M<\/jats:italic>-divisibility of stable laws.<\/jats:p>","DOI":"10.1017\/s0305004100053925","type":"journal-article","created":{"date-parts":[[2008,11,7]],"date-time":"2008-11-07T15:17:41Z","timestamp":1226071061000},"page":"289-295","source":"Crossref","is-referenced-by-count":24,"title":["Some further results in infinite divisibility"],"prefix":"10.1017","volume":"82","author":[{"given":"D. N.","family":"Shanbhag","sequence":"first","affiliation":[]},{"given":"D.","family":"Pestana","sequence":"additional","affiliation":[]},{"given":"M.","family":"Sreehari","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2008,10,24]]},"reference":[{"key":"S0305004100053925_ref012","doi-asserted-by":"publisher","DOI":"10.1137\/1112062"},{"key":"S0305004100053925_ref011","doi-asserted-by":"publisher","DOI":"10.1093\/biomet\/64.1.167"},{"key":"S0305004100053925_ref010","volume-title":"A course of modern analysis","author":"Whittaker","year":"1963"},{"key":"S0305004100053925_ref008","volume-title":"Preservation of infinite divisibility under mixing and related topics","author":"Steutel","year":"1970"},{"key":"S0305004100053925_ref006","doi-asserted-by":"publisher","DOI":"10.1214\/aoms\/1177696912"},{"key":"S0305004100053925_ref004","doi-asserted-by":"publisher","DOI":"10.1214\/aoms\/1177693436"},{"key":"S0305004100053925_ref003","doi-asserted-by":"publisher","DOI":"10.1214\/aop\/1176996756"},{"key":"S0305004100053925_ref002","doi-asserted-by":"publisher","DOI":"10.1017\/S0305004100042225"},{"key":"S0305004100053925_ref005","volume-title":"Characteristic functions","author":"Lukacs","year":"1970"},{"key":"S0305004100053925_ref009","doi-asserted-by":"publisher","DOI":"10.1016\/0304-4149(73)90008-2"},{"key":"S0305004100053925_ref007","doi-asserted-by":"publisher","DOI":"10.1007\/BF00537265"},{"key":"S0305004100053925_ref001","volume-title":"Higher transcendental functions","volume":"I","author":"Erd\u00e9lyi","year":"1953"}],"container-title":["Mathematical Proceedings of the Cambridge Philosophical Society"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0305004100053925","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,27]],"date-time":"2019-05-27T21:10:20Z","timestamp":1558991420000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0305004100053925\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1977,9]]},"references-count":12,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1977,9]]}},"alternative-id":["S0305004100053925"],"URL":"https:\/\/doi.org\/10.1017\/s0305004100053925","relation":{},"ISSN":["0305-0041","1469-8064"],"issn-type":[{"value":"0305-0041","type":"print"},{"value":"1469-8064","type":"electronic"}],"subject":[],"published":{"date-parts":[[1977,9]]}}}