{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,4,27]],"date-time":"2023-04-27T11:10:03Z","timestamp":1682593803590},"reference-count":9,"publisher":"Cambridge University Press (CUP)","issue":"01","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. Appl. Probab."],"published-print":{"date-parts":[[2014,3]]},"abstract":"<jats:p>We consider a one-dimensional discrete symmetric random walk with a reflecting boundary at the origin. Generating functions are found for the two-dimensional probability distribution P{<jats:italic>S<\/jats:italic>\n               <jats:sub>\n                  <jats:italic>n<\/jats:italic>\n               <\/jats:sub> = <jats:italic>x<\/jats:italic>, max<jats:sub>1\u2264<jats:italic>j<\/jats:italic>\u2264<jats:italic>n<\/jats:italic>\n               <\/jats:sub>\n               <jats:italic>S<\/jats:italic>\n               <jats:sub>\n                  <jats:italic>n<\/jats:italic>\n               <\/jats:sub> = <jats:italic>a<\/jats:italic>} of being at position <jats:italic>x<\/jats:italic> after <jats:italic>n<\/jats:italic> steps, while the maximal location that the walker has achieved during these <jats:italic>n<\/jats:italic> steps is <jats:italic>a<\/jats:italic>. We also obtain the familiar (marginal) one-dimensional distribution for <jats:italic>S<\/jats:italic>\n               <jats:sub>\n                  <jats:italic>n<\/jats:italic>\n               <\/jats:sub> = <jats:italic>x<\/jats:italic>, but more importantly that for max<jats:sub>1\u2264<jats:italic>j<\/jats:italic>\u2264<jats:italic>n<\/jats:italic>\n               <\/jats:sub>\n               <jats:italic>S<\/jats:italic>\n               <jats:sub>\n                  <jats:italic>j<\/jats:italic>\n               <\/jats:sub> = <jats:italic>a<\/jats:italic> asymptotically at fixed <jats:italic>a<\/jats:italic>\n               <jats:sup>2<\/jats:sup> \/ <jats:italic>n<\/jats:italic>. We are able to compute and compare the expectations and variances of the two one-dimensional distributions, finding that they have qualitatively similar forms, but differ quantitatively in the anticipated fashion.<\/jats:p>","DOI":"10.1017\/s0021900200010159","type":"journal-article","created":{"date-parts":[[2016,3,29]],"date-time":"2016-03-29T14:49:37Z","timestamp":1459262977000},"page":"162-173","source":"Crossref","is-referenced-by-count":0,"title":["The Maximum of a Symmetric Next Neighbor Walk on the Nonnegative Integers"],"prefix":"10.1017","volume":"51","author":[{"given":"Ora E.","family":"Percus","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jerome K.","family":"Percus","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2016,2,19]]},"reference":[{"key":"S0021900200010159_ref9","volume-title":"Stochastic Processes: Problems and Solutions","year":"1960"},{"key":"S0021900200010159_ref8","doi-asserted-by":"publisher","DOI":"10.2307\/1427121"},{"key":"S0021900200010159_ref7","volume-title":"Mathematics of Genome Analysis","year":"2002"},{"key":"S0021900200010159_ref6","volume-title":"Real Analysis: Measures, Integrals and Applications","year":"2013"},{"key":"S0021900200010159_ref1","doi-asserted-by":"publisher","DOI":"10.1214\/aoms\/1177699147"},{"key":"S0021900200010159_ref4","volume-title":"Tables of Integrals, Series, and Products","year":"1965"},{"key":"S0021900200010159_ref3","volume-title":"An Introduction to Probability Theory and Its Applications","volume":"I","year":"1968"},{"key":"S0021900200010159_ref2","volume-title":"Tables of Integral Transforms","volume":"I","year":"1954"},{"key":"S0021900200010159_ref5","doi-asserted-by":"publisher","DOI":"10.2307\/2304386"}],"container-title":["Journal of Applied Probability"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0021900200010159","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,4,26]],"date-time":"2023-04-26T08:08:44Z","timestamp":1682496524000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0021900200010159\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,3]]},"references-count":9,"journal-issue":{"issue":"01","published-print":{"date-parts":[[2014,3]]}},"alternative-id":["S0021900200010159"],"URL":"https:\/\/doi.org\/10.1017\/s0021900200010159","relation":{},"ISSN":["0021-9002","1475-6072"],"issn-type":[{"value":"0021-9002","type":"print"},{"value":"1475-6072","type":"electronic"}],"subject":[],"published":{"date-parts":[[2014,3]]}}}