{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,4,26]],"date-time":"2023-04-26T08:40:30Z","timestamp":1682498430432},"reference-count":23,"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":"Let {Z<\/jats:italic>\n \n n<\/jats:italic>\n <\/jats:sub>}\n n<\/jats:italic>\u22650<\/jats:sub> be a random walk with a negative drift and independent and identically distributed increments with heavy-tailed distribution, and let M<\/jats:italic> = sup\n n<\/jats:italic>\u22650<\/jats:sub>\n Z<\/jats:italic>\n \n n<\/jats:italic>\n <\/jats:sub> be its supremum. Asmussen and Kl\u00fcppelberg (1996) considered the behavior of the random walk given that M<\/jats:italic> &gt; x<\/jats:italic> for large x<\/jats:italic>, and obtained a limit theorem, as x<\/jats:italic> \u2192 \u221e, for the distribution of the quadruple that includes the time \u03c4 = \u03c4(x<\/jats:italic>) to exceed level x<\/jats:italic>, position Z<\/jats:italic>\n \u03c4<\/jats:sub> at this time, position Z<\/jats:italic>\n \u03c4-1<\/jats:sub> at the prior time, and the trajectory up to it (similar results were obtained for the Cram\u00e9r-Lundberg insurance risk process). We obtain here several extensions of this result to various regenerative-type models and, in particular, to the case of a random walk with dependent increments. Particular attention is given to describing the limiting conditional behavior of \u03c4. The class of models includes Markov-modulated models as particular cases. We also study fluid models, the Bj\u00f6rk-Grandell risk process, give examples where the order of \u03c4 is genuinely different from the random walk case, and discuss which growth rates are possible. Our proofs are purely probabilistic and are based on results and ideas from Asmussen, Schmidli and Schmidt (1999), Foss and Zachary (2002), and Foss, Konstantopoulos and Zachary (2007).<\/jats:p>","DOI":"10.1017\/s0021900200010135","type":"journal-article","created":{"date-parts":[[2016,3,29]],"date-time":"2016-03-29T14:49:37Z","timestamp":1459262977000},"page":"136-151","source":"Crossref","is-referenced-by-count":5,"title":["On Exceedance Times for Some Processes with Dependent Increments"],"prefix":"10.1017","volume":"51","author":[{"given":"S\u00f8ren","family":"Asmussen","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Sergey","family":"Foss","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2016,2,19]]},"reference":[{"key":"S0021900200010135_ref19","first-page":"37","volume-title":"Analytic Methods in Applied Probability","year":"2002"},{"key":"S0021900200010135_ref20","doi-asserted-by":"publisher","DOI":"10.1007\/s10959-007-0081-2"},{"key":"S0021900200010135_ref17","doi-asserted-by":"publisher","DOI":"10.3150\/10-BEJ251"},{"key":"S0021900200010135_ref16","doi-asserted-by":"publisher","DOI":"10.1239\/jap\/1197908822"},{"key":"S0021900200010135_ref15","doi-asserted-by":"publisher","DOI":"10.1016\/0304-4149(94)90113-9"},{"key":"S0021900200010135_ref9","doi-asserted-by":"publisher","DOI":"10.1214\/105051604000000044"},{"key":"S0021900200010135_ref14","doi-asserted-by":"publisher","DOI":"10.1137\/1110037"},{"key":"S0021900200010135_ref8","doi-asserted-by":"publisher","DOI":"10.1239\/aap\/1029955142"},{"key":"S0021900200010135_ref13","doi-asserted-by":"publisher","DOI":"10.1080\/03461238.1988.10413839"},{"key":"S0021900200010135_ref7","doi-asserted-by":"publisher","DOI":"10.1016\/S0304-4149(98)00064-7"},{"key":"S0021900200010135_ref12","volume-title":"Regular Variation","year":"1987"},{"key":"S0021900200010135_ref6","first-page":"591","volume":"3","year":"2011","journal-title":"Stoch. 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