{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,4,26]],"date-time":"2023-04-26T07:41:08Z","timestamp":1682494868766},"reference-count":13,"publisher":"Cambridge University Press (CUP)","issue":"03","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. Appl. Probab."],"published-print":{"date-parts":[[2012,9]]},"abstract":"<jats:p>An infinite sequence (<jats:italic>Y<\/jats:italic>\n               <jats:sub>1<\/jats:sub>, <jats:italic>Y<\/jats:italic>\n               <jats:sub>2<\/jats:sub>,\u2026) of independent Bernoulli random variables with P(<jats:italic>Y<\/jats:italic>\n               <jats:sub>\n                  <jats:italic>i<\/jats:italic>\n               <\/jats:sub> = 1) = <jats:italic>a<\/jats:italic> \/ (<jats:italic>a<\/jats:italic> + <jats:italic>b<\/jats:italic> + <jats:italic>i<\/jats:italic> - 1), <jats:italic>i<\/jats:italic> = 1, 2,\u2026, where <jats:italic>a<\/jats:italic> &amp;gt; 0 and <jats:italic>b<\/jats:italic> \u2265 0, will be called a Bern(<jats:italic>a<\/jats:italic>, <jats:italic>b<\/jats:italic>) sequence. Consider the counts <jats:italic>Z<\/jats:italic>\n               <jats:sub>1<\/jats:sub>, <jats:italic>Z<\/jats:italic>\n               <jats:sub>2<\/jats:sub>, <jats:italic>Z<\/jats:italic>\n               <jats:sub>3<\/jats:sub>,\u2026 of occurrences of patterns or strings of the form {11}, {101}, {1001},\u2026, respectively, in this sequence. The joint distribution of the counts <jats:italic>Z<\/jats:italic>\n               <jats:sub>1<\/jats:sub>, <jats:italic>Z<\/jats:italic>\n               <jats:sub>2<\/jats:sub>,\u2026 in the infinite Bern(<jats:italic>a<\/jats:italic>, <jats:italic>b<\/jats:italic>) sequence has been studied extensively. The counts from the initial finite sequence (<jats:italic>Y<\/jats:italic>\n               <jats:sub>1<\/jats:sub>, <jats:italic>Y<\/jats:italic>\n               <jats:sub>2<\/jats:sub>,\u2026, <jats:italic>Y<\/jats:italic>\n               <jats:sub>\n                  <jats:italic>n<\/jats:italic>\n               <\/jats:sub>) have been studied by Holst (2007), (2008b), who obtained the joint factorial moments for Bern(<jats:italic>a<\/jats:italic>, 0) and the factorial moments of <jats:italic>Z<\/jats:italic>\n               <jats:sub>1<\/jats:sub>, the count of the string {1, 1}, for a general Bern(<jats:italic>a<\/jats:italic>, <jats:italic>b<\/jats:italic>). We consider stopping the Bernoulli sequence at a random time and describe the joint distribution of counts, which extends Holst's results. We show that the joint distribution of counts from a class of randomly stopped Bernoulli sequences possesses the <jats:italic>mixture of independent Poissons<\/jats:italic> property: there is a random vector conditioned on which the counts are independent Poissons. To obtain these results, we extend the <jats:italic>conditional marked Poisson process<\/jats:italic> technique introduced in Huffer, Sethuraman and Sethuraman (2009). Our results avoid previous combinatorial and induction methods which generally only yield factorial moments.<\/jats:p>","DOI":"10.1017\/s0021900200009529","type":"journal-article","created":{"date-parts":[[2016,3,29]],"date-time":"2016-03-29T14:49:46Z","timestamp":1459262986000},"page":"758-772","source":"Crossref","is-referenced-by-count":1,"title":["Joint Distributions of Counts of Strings in Finite Bernoulli Sequences"],"prefix":"10.1017","volume":"49","author":[{"given":"Fred W.","family":"Huffer","sequence":"first","affiliation":[]},{"given":"Jayaram","family":"Sethuraman","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2016,2,4]]},"reference":[{"key":"S0021900200009529_ref7","doi-asserted-by":"publisher","DOI":"10.1239\/jap\/1261670698"},{"key":"S0021900200009529_ref6","doi-asserted-by":"publisher","DOI":"10.1239\/jap\/1222441836"},{"key":"S0021900200009529_ref5","doi-asserted-by":"publisher","DOI":"10.1239\/jap\/1231340241"},{"key":"S0021900200009529_ref4","doi-asserted-by":"publisher","DOI":"10.1239\/jap\/1189717547"},{"key":"S0021900200009529_ref8","doi-asserted-by":"publisher","DOI":"10.1007\/s11512-010-0131-3"},{"key":"S0021900200009529_ref15","doi-asserted-by":"publisher","DOI":"10.1214\/lnms\/1196285386"},{"key":"S0021900200009529_ref10","first-page":"2125","volume":"137","year":"2009","journal-title":"Proc. Amer. Math. Soc."},{"key":"S0021900200009529_ref14","volume-title":"Adventures in Stochastic Processes","year":"1992"},{"key":"S0021900200009529_ref1","doi-asserted-by":"publisher","DOI":"10.1214\/aoap\/1177005647"},{"key":"S0021900200009529_ref13","doi-asserted-by":"publisher","DOI":"10.1137\/1116005"},{"key":"S0021900200009529_ref12","volume-title":"Poisson Processes","year":"1993"},{"key":"S0021900200009529_ref11","doi-asserted-by":"publisher","DOI":"10.1023\/B:JOTP.0000020485.34082.8c"},{"key":"S0021900200009529_ref2","volume-title":"Logarithmic Combinatorial Structures: A Probabilistic Approach","year":"2003"}],"container-title":["Journal of Applied Probability"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0021900200009529","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,4,26]],"date-time":"2023-04-26T07:12:38Z","timestamp":1682493158000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0021900200009529\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,9]]},"references-count":13,"journal-issue":{"issue":"03","published-print":{"date-parts":[[2012,9]]}},"alternative-id":["S0021900200009529"],"URL":"https:\/\/doi.org\/10.1017\/s0021900200009529","relation":{},"ISSN":["0021-9002","1475-6072"],"issn-type":[{"value":"0021-9002","type":"print"},{"value":"1475-6072","type":"electronic"}],"subject":[],"published":{"date-parts":[[2012,9]]}}}