{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,7]],"date-time":"2026-04-07T02:06:27Z","timestamp":1775527587189,"version":"3.50.1"},"reference-count":38,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2025,11,21]],"date-time":"2025-11-21T00:00:00Z","timestamp":1763683200000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0\/"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["J. Appl. Probab."],"published-print":{"date-parts":[[2026,3]]},"abstract":"Abstract<\/jats:title>\n \n Let\n \n \n \n $\\{X_{i}\\}_{i\\geq1}$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n be a sequence of independent and identically distributed random variables and\n \n \n \n $T\\in\\{1,2,\\ldots\\}$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n a stopping time associated with this sequence. In this paper, the distribution of the minimum observation,\n \n \n \n $\\min\\{X_{1},X_{2},\\ldots,X_{T}\\}$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , until the stopping time\n T<\/jats:italic>\n is provided by proposing a methodology based on an appropriate change of the initial probability measure of the probability space to a truncated (shifted) one on the\n \n \n \n $X_{i}$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . As an application of the aforementioned general result, the random variables\n \n \n \n $X_{1},X_{2},\\ldots$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n are considered to be the interarrival times (spacings) between successive appearances of events in a renewal counting process\n \n \n \n $\\{Y_{t},t\\geq0\\}$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , while the stopping time\n T<\/jats:italic>\n is set to be the number of summands until the sum of the\n \n \n \n $X_{i}$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n exceeds\n t<\/jats:italic>\n for the first time, i.e.\n \n \n \n $T=Y_{t}+1$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . Under this setup, the distribution of the minimal spacing,\n \n \n \n $D_{t}=\\min\\{X_{1},X_{2},\\ldots,X_{Y_{t}+1}\\}$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , that starts in the interval [0,\n t<\/jats:italic>\n ] is investigated and a stochastic ordering relation for\n \n \n \n $D_{t}$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n is obtained. In addition, bounds for the tail probability of\n \n \n \n $D_{t}$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n are provided when the interarrival times have the increasing failure rate \/ decreasing failure rate property. In the special case of a Poisson process, an exact formula, as well as closed-form bounds and an asymptotic result, are derived for the tail probability of\n \n \n \n $D_{t}$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . Furthermore, for renewal processes with Erlang and uniformly distributed interarrival times, exact and approximation formulae for the tail probability of\n \n \n \n $D_{t}$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n are also proposed. Finally, numerical examples are presented to illustrate the aforementioned exact and asymptotic results, and practical applications are briefly discussed.\n <\/jats:p>","DOI":"10.1017\/jpr.2025.10026","type":"journal-article","created":{"date-parts":[[2025,11,21]],"date-time":"2025-11-21T05:31:31Z","timestamp":1763703091000},"page":"157-176","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":0,"title":["The distribution of the minimum observation until a stopping time, with an application to the minimal spacing in a Renewal process"],"prefix":"10.1017","volume":"63","author":[{"given":"Eutichia","family":"Vaggelatou","sequence":"first","affiliation":[{"name":"National and Kapodistrian University of Athens"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2025,11,21]]},"reference":[{"key":"S0021900225100260_ref30","first-page":"124194","article-title":"General large deviations of longest gaps in homogeneous Poisson processes","volume":"489","author":"Omwonylee","year":"2020","journal-title":"J. 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Springer, New York.","DOI":"10.1007\/978-1-4612-2528-7"},{"key":"S0021900225100260_ref33","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-387-34675-5"},{"key":"S0021900225100260_ref16","doi-asserted-by":"publisher","DOI":"10.1214\/aoms\/1177729030"}],"container-title":["Journal of Applied Probability"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0021900225100260","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,7]],"date-time":"2026-04-07T01:25:27Z","timestamp":1775525127000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0021900225100260\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,11,21]]},"references-count":38,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2026,3]]}},"alternative-id":["S0021900225100260"],"URL":"https:\/\/doi.org\/10.1017\/jpr.2025.10026","relation":{},"ISSN":["0021-9002","1475-6072"],"issn-type":[{"value":"0021-9002","type":"print"},{"value":"1475-6072","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,11,21]]},"assertion":[{"value":"\u00a9 The Author(s), 2025. Published by Cambridge University Press on behalf of Applied Probability Trust","name":"copyright","label":"Copyright","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}},{"value":"This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0\/), which permits re-use and distribution in any medium or format in unadapted form only, for noncommercial purposes only, provided the original work is properly cited.","name":"license","label":"License","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}},{"value":"This content has been made available to all.","name":"free","label":"Free to read"}]}}