{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,4,26]],"date-time":"2023-04-26T09:41:05Z","timestamp":1682502065310},"reference-count":4,"publisher":"Cambridge University Press (CUP)","issue":"02","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. Appl. Probab."],"published-print":{"date-parts":[[2014,6]]},"abstract":"<jats:p>Let <jats:italic>X<\/jats:italic>, <jats:italic>B<\/jats:italic>, and <jats:italic>Y<\/jats:italic> be the Dirichlet, Bernoulli, and beta-independent random variables such that <jats:italic>X<\/jats:italic> ~ <jats:bold>\n                  <jats:italic>D<\/jats:italic>\n               <\/jats:bold>(<jats:italic>a<\/jats:italic>\n               <jats:sub>0<\/jats:sub>, \u2026, <jats:italic>a<\/jats:italic>\n               <jats:sub>\n                  <jats:italic>d<\/jats:italic>\n               <\/jats:sub>), Pr(<jats:italic>B<\/jats:italic> = (0, \u2026, 0, 1, 0, \u2026, 0)) = <jats:italic>a<\/jats:italic>\n               <jats:sub>\n                  <jats:italic>i<\/jats:italic>\n               <\/jats:sub> \/ <jats:italic>a<\/jats:italic> with <jats:italic>a<\/jats:italic> = \u2211<jats:sub>\n                  <jats:italic>i<\/jats:italic>=0<\/jats:sub>\n               <jats:sup>\n                  <jats:italic>d<\/jats:italic>\n               <\/jats:sup>\n               <jats:italic>a<\/jats:italic>\n               <jats:sub>\n                  <jats:italic>i<\/jats:italic>\n               <\/jats:sub>, and <jats:italic>Y<\/jats:italic> ~ \u03b2(1, <jats:italic>a<\/jats:italic>). Then, as proved by Sethuraman (1994), <jats:italic>X<\/jats:italic> ~ <jats:italic>X<\/jats:italic>(1 - <jats:italic>Y<\/jats:italic>) + <jats:italic>BY<\/jats:italic>. This gives the stationary distribution of a simple Markov chain on a tetrahedron. In this paper we introduce a new distribution on the tetrahedron called a quasi-Bernoulli distribution <jats:bold>B<\/jats:bold>\n               <jats:sub>\n                  <jats:italic>k<\/jats:italic>\n               <\/jats:sub>(<jats:italic>a<\/jats:italic>\n               <jats:sub>0<\/jats:sub>, \u2026, <jats:italic>a<\/jats:italic>\n               <jats:sub>\n                  <jats:italic>d<\/jats:italic>\n               <\/jats:sub>) with <jats:italic>k<\/jats:italic> an integer such that the above result holds when <jats:italic>B<\/jats:italic> follows <jats:bold>B<\/jats:bold>\n               <jats:sub>\n                  <jats:italic>k<\/jats:italic>\n               <\/jats:sub>(<jats:italic>a<\/jats:italic>\n               <jats:sub>0<\/jats:sub>, \u2026, <jats:italic>a<\/jats:italic>\n               <jats:sub>\n                  <jats:italic>d<\/jats:italic>\n               <\/jats:sub>) and when <jats:italic>Y<\/jats:italic> ~ \u03b2(<jats:italic>k<\/jats:italic>, <jats:italic>a<\/jats:italic>). We extend it even more generally to the case where <jats:italic>X<\/jats:italic> and <jats:italic>B<\/jats:italic> are random probabilities such that <jats:italic>X<\/jats:italic> is Dirichlet and <jats:italic>B<\/jats:italic> is quasi-Bernoulli. Finally, the case where the integer <jats:italic>k<\/jats:italic> is replaced by a positive number <jats:italic>c<\/jats:italic> is considered when <jats:italic>a<\/jats:italic>\n               <jats:sub>0<\/jats:sub> = \u00b7 \u00b7 \u00b7 = <jats:italic>a<\/jats:italic>\n               <jats:sub>\n                  <jats:italic>d<\/jats:italic>\n               <\/jats:sub> = 1.<\/jats:p>","DOI":"10.1017\/s0001867800011320","type":"journal-article","created":{"date-parts":[[2016,3,29]],"date-time":"2016-03-29T14:51:11Z","timestamp":1459263071000},"page":"400-416","source":"Crossref","is-referenced-by-count":0,"title":["Dirichlet and Quasi-Bernoulli Laws for Perpetuities"],"prefix":"10.1017","volume":"51","author":[{"given":"Pawe\u0142","family":"Hitczenko","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"G\u00e9rard","family":"Letac","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2016,2,19]]},"reference":[{"key":"S0001867800011320_ref8","volume-title":"Continuous Multivariate Distributions","volume":"1","year":"2000"},{"key":"S0001867800011320_ref11","first-page":"639","volume":"4","year":"1994","journal-title":"Statistica Sinica"},{"key":"S0001867800011320_ref2","first-page":"161","volume":"4","year":"2010","journal-title":"Commun. Stoch. Anal."},{"key":"S0001867800011320_ref5","first-page":"97","volume-title":"Bayesian Statistics 5","year":"1996"}],"container-title":["Journal of Applied Probability"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0001867800011320","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,4,26]],"date-time":"2023-04-26T09:06:41Z","timestamp":1682500001000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0001867800011320\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,6]]},"references-count":4,"journal-issue":{"issue":"02","published-print":{"date-parts":[[2014,6]]}},"alternative-id":["S0001867800011320"],"URL":"https:\/\/doi.org\/10.1017\/s0001867800011320","relation":{},"ISSN":["0021-9002","1475-6072"],"issn-type":[{"value":"0021-9002","type":"print"},{"value":"1475-6072","type":"electronic"}],"subject":[],"published":{"date-parts":[[2014,6]]}}}