{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,25]],"date-time":"2025-02-25T13:48:31Z","timestamp":1740491311468,"version":"3.38.0"},"reference-count":3,"publisher":"Cambridge University Press (CUP)","issue":"03","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. Appl. Probab."],"published-print":{"date-parts":[[2014,9]]},"abstract":"<jats:p>Suppose that <jats:italic>X<\/jats:italic>\n               <jats:sub>1<\/jats:sub>, \u2026, <jats:italic>X<\/jats:italic>\n               <jats:sub>\n                  <jats:italic>n<\/jats:italic>\n               <\/jats:sub> are random variables with the same known marginal distribution <jats:italic>F<\/jats:italic> but unknown dependence structure. In this paper we study the smallest possible value of <jats:bold>P<\/jats:bold>(<jats:italic>X<\/jats:italic>\n               <jats:sub>1<\/jats:sub> + \u00b7 \u00b7 \u00b7 + <jats:italic>X<\/jats:italic>\n               <jats:sub>\n                  <jats:italic>n<\/jats:italic>\n               <\/jats:sub> &amp;lt; <jats:italic>s<\/jats:italic>) over all possible dependence structures, denoted by <jats:italic>m<\/jats:italic>\n               <jats:sub>\n                  <jats:italic>n<\/jats:italic>,<jats:italic>F<\/jats:italic>\n               <\/jats:sub>(<jats:italic>s<\/jats:italic>). We show that <jats:italic>m<\/jats:italic>\n               <jats:sub>\n                  <jats:italic>n<\/jats:italic>,<jats:italic>F<\/jats:italic>\n               <\/jats:sub>(<jats:italic>ns<\/jats:italic>) \u2192 0 for <jats:italic>s<\/jats:italic> no more than the mean of <jats:italic>F<\/jats:italic> under weak assumptions. We also derive a limit of <jats:italic>m<\/jats:italic>\n               <jats:sub>\n                  <jats:italic>n<\/jats:italic>,<jats:italic>F<\/jats:italic>\n               <\/jats:sub>(<jats:italic>ns<\/jats:italic>) for any <jats:italic>s<\/jats:italic> \u2208 <jats:bold>R<\/jats:bold> with an error of at most <jats:italic>n<\/jats:italic>\n               <jats:sup>-1\/6<\/jats:sup> for general continuous distributions. An application of our result to risk management confirms that the worst-case value at risk is asymptotically equivalent to the worst-case expected shortfall for risk aggregation with dependence uncertainty. In the last part of this paper we present a dual presentation of the theory of complete mixability and give dual proofs of theorems in the literature on this concept.<\/jats:p>","DOI":"10.1017\/s0001867800011678","type":"journal-article","created":{"date-parts":[[2016,3,29]],"date-time":"2016-03-29T10:50:01Z","timestamp":1459248601000},"page":"780-798","source":"Crossref","is-referenced-by-count":2,"title":["Asymptotic Bounds for the Distribution of the Sum of Dependent Random Variables"],"prefix":"10.1017","volume":"51","author":[{"given":"Ruodu","family":"Wang","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2016,2,4]]},"reference":[{"key":"S0001867800011678_ref10","first-page":"1","volume":"16","year":"2014","journal-title":"J. Risk"},{"first-page":"111","volume-title":"Copula Theory and Its Applications","year":"2010","key":"S0001867800011678_ref5"},{"volume-title":"Mathematical Risk Analysis. Dependence, Risk Bounds, Optimal Allocations and Portfolios","year":"2013","key":"S0001867800011678_ref14"}],"container-title":["Journal of Applied Probability"],"original-title":[],"language":"en","deposited":{"date-parts":[[2017,2,7]],"date-time":"2017-02-07T23:20:31Z","timestamp":1486509631000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0001867800011678\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,9]]},"references-count":3,"journal-issue":{"issue":"03","published-print":{"date-parts":[[2014,9]]}},"alternative-id":["S0001867800011678"],"URL":"https:\/\/doi.org\/10.1017\/s0001867800011678","relation":{},"ISSN":["0021-9002","1475-6072"],"issn-type":[{"type":"print","value":"0021-9002"},{"type":"electronic","value":"1475-6072"}],"subject":[],"published":{"date-parts":[[2014,9]]}}}