{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,25]],"date-time":"2025-02-25T13:48:27Z","timestamp":1740491307856,"version":"3.38.0"},"reference-count":15,"publisher":"Cambridge University Press (CUP)","issue":"01","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. Appl. Probab."],"published-print":{"date-parts":[[2015,3]]},"abstract":"<jats:p>Let <jats:italic>\n                  <jats:bold>W<\/jats:bold>\n               <\/jats:italic> = {<jats:italic>\n                  <jats:bold>W<\/jats:bold>\n               <\/jats:italic>\n               <jats:sub>\n                  <jats:italic>n<\/jats:italic>\n               <\/jats:sub>: <jats:italic>n<\/jats:italic> \u2208 <jats:bold>N<\/jats:bold>} be a sequence of random vectors in <jats:bold>R<\/jats:bold>\n               <jats:sup>\n                  <jats:italic>d<\/jats:italic>\n               <\/jats:sup>, <jats:italic>d<\/jats:italic> \u2265 1. In this paper we consider the logarithmic asymptotics of the extremes of <jats:italic>\n                  <jats:bold>W<\/jats:bold>\n               <\/jats:italic>, that is, for any vector <jats:italic>\n                  <jats:bold>q<\/jats:bold>\n               <\/jats:italic> &amp;gt; 0 in <jats:bold>R<\/jats:bold>\n               <jats:sup>\n                  <jats:italic>d<\/jats:italic>\n               <\/jats:sup>, we find that log<jats:bold>P<\/jats:bold>(there exists <jats:italic>n<\/jats:italic> \u2208 <jats:bold>N<\/jats:bold>: <jats:italic>\n                  <jats:bold>W<\/jats:bold>\n               <\/jats:italic>\n               <jats:sub>\n                  <jats:italic>n<\/jats:italic>\n               <\/jats:sub> \n               <jats:italic>u<\/jats:italic> \n               <jats:italic>\n                  <jats:bold>q<\/jats:bold>\n               <\/jats:italic>) as <jats:italic>u<\/jats:italic> \u2192 \u221e. We follow the approach of the restricted large deviation principle introduced in Duffy (2003). That is, we assume that, for every <jats:italic>\n                  <jats:bold>q<\/jats:bold>\n               <\/jats:italic> \u2265 <jats:bold>0<\/jats:bold>, and some scalings {<jats:italic>a<\/jats:italic>\n               <jats:sub>\n                  <jats:italic>n<\/jats:italic>\n               <\/jats:sub>}, {<jats:italic>v<\/jats:italic>\n               <jats:sub>\n                  <jats:italic>n<\/jats:italic>\n               <\/jats:sub>}, (1 \/ <jats:italic>v<\/jats:italic>\n               <jats:sub>\n                  <jats:italic>n<\/jats:italic>\n               <\/jats:sub>)log<jats:bold>P<\/jats:bold>(<jats:italic>\n                  <jats:bold>W<\/jats:bold>\n               <\/jats:italic>\n               <jats:sub>\n                  <jats:italic>n<\/jats:italic>\n               <\/jats:sub> \/\n<jats:italic>a<\/jats:italic>\n               <jats:sub>\n                  <jats:italic>n<\/jats:italic>\n               <\/jats:sub> \u2265 <jats:italic>u<\/jats:italic> \n               <jats:italic>\n                  <jats:bold>q<\/jats:bold>\n               <\/jats:italic>) has a, continuous in <jats:italic>\n                  <jats:bold>q<\/jats:bold>\n               <\/jats:italic>, limit <jats:italic>J<\/jats:italic>\n               <jats:sub>\n                  <jats:italic>\n                     <jats:bold>W<\/jats:bold>\n                  <\/jats:italic>\n               <\/jats:sub>(<jats:italic>\n                  <jats:bold>q<\/jats:bold>\n               <\/jats:italic>). We allow the scalings {<jats:italic>a<\/jats:italic>\n               <jats:sub>\n                  <jats:italic>n<\/jats:italic>\n               <\/jats:sub>} and {<jats:italic>v<\/jats:italic>\n               <jats:sub>\n                  <jats:italic>n<\/jats:italic>\n               <\/jats:sub>} to be regularly varying with a positive index. This approach is general enough to incorporate sequences <jats:italic>\n                  <jats:bold>W<\/jats:bold>\n               <\/jats:italic>, such that the probability law of <jats:italic>\n                  <jats:bold>W<\/jats:bold>\n               <\/jats:italic>\n               <jats:sub>\n                  <jats:italic>n<\/jats:italic>\n               <\/jats:sub> \/ <jats:italic>a<\/jats:italic>\n               <jats:sub>\n                  <jats:italic>n<\/jats:italic>\n               <\/jats:sub> satisfies the large deviation principle with continuous, not necessarily convex, rate functions. The equations for these asymptotics are in agreement with the literature.<\/jats:p>","DOI":"10.1017\/s0021900200012201","type":"journal-article","created":{"date-parts":[[2016,3,29]],"date-time":"2016-03-29T10:49:27Z","timestamp":1459248567000},"page":"68-81","source":"Crossref","is-referenced-by-count":1,"title":["Logarithmic Asymptotics for Multidimensional Extremes Under Nonlinear Scalings"],"prefix":"10.1017","volume":"52","author":[{"given":"K. M.","family":"Kosi\u0144ski","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"M.","family":"Mandjes","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2016,2,4]]},"reference":[{"key":"S0021900200012201_ref9","first-page":"131","volume-title":"Studies in Applied Probability","volume":"31A","year":"1994"},{"key":"S0021900200012201_ref10","doi-asserted-by":"publisher","DOI":"10.1109\/90.251894"},{"key":"S0021900200012201_ref7","doi-asserted-by":"publisher","DOI":"10.1239\/jap\/1208358955"},{"key":"S0021900200012201_ref6","doi-asserted-by":"publisher","DOI":"10.1017\/S0305004100073709"},{"key":"S0021900200012201_ref5","doi-asserted-by":"publisher","DOI":"10.1016\/j.spa.2010.08.010"},{"volume-title":"Large Deviations Techniques and Applications","year":"1998","key":"S0021900200012201_ref4"},{"key":"S0021900200012201_ref3","doi-asserted-by":"publisher","DOI":"10.1214\/aop\/1041903218"},{"volume-title":"Convex Analysis","year":"1970","key":"S0021900200012201_ref15"},{"volume-title":"Regular Variation","year":"1987","key":"S0021900200012201_ref2"},{"key":"S0021900200012201_ref14","doi-asserted-by":"publisher","DOI":"10.1214\/aoms\/1177706632"},{"volume-title":"An Introduction to Continuity, Extrema, and Related Topics for General Gaussian Processes","year":"1990","key":"S0021900200012201_ref1"},{"key":"S0021900200012201_ref13","first-page":"229","volume":"2","year":"1995","journal-title":"J. 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