{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,12]],"date-time":"2026-05-12T03:40:09Z","timestamp":1778557209982,"version":"3.51.4"},"reference-count":34,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2018,1,30]],"date-time":"2018-01-30T00:00:00Z","timestamp":1517270400000},"content-version":"unspecified","delay-in-days":1521,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Journal of Applied Probability"],"published-print":{"date-parts":[[2013,12]]},"abstract":"<jats:p>In this paper we study the fractional moments of the stationary solution to the stochastic recurrence equation<jats:italic>X<\/jats:italic><jats:sub><jats:italic>t<\/jats:italic><\/jats:sub>=<jats:italic>A<\/jats:italic><jats:sub><jats:italic>t<\/jats:italic><\/jats:sub><jats:italic>X<\/jats:italic><jats:sub><jats:italic>t<\/jats:italic>\u22121<\/jats:sub>+<jats:italic>B<\/jats:italic><jats:sub><jats:italic>t<\/jats:italic><\/jats:sub>,<jats:italic>t<\/jats:italic>\u2208<jats:bold>Z<\/jats:bold>, where ((<jats:italic>A<\/jats:italic><jats:sub><jats:italic>t<\/jats:italic><\/jats:sub>,<jats:italic>B<\/jats:italic><jats:sub><jats:italic>t<\/jats:italic><\/jats:sub>))<jats:sub><jats:italic>t<\/jats:italic>\u2208<jats:bold>Z<\/jats:bold><\/jats:sub>is an independent and identically distributed bivariate sequence. We derive recursive formulae for the fractional moments E|<jats:italic>X<\/jats:italic><jats:sub>0<\/jats:sub>|<jats:sup><jats:italic>p<\/jats:italic><\/jats:sup>,<jats:italic>p<\/jats:italic>\u2208<jats:bold>R<\/jats:bold>. Special attention is given to the case when<jats:italic>B<\/jats:italic><jats:sub><jats:italic>t<\/jats:italic><\/jats:sub>has an Erlang distribution. We provide various approximations to the moments E|<jats:italic>X<\/jats:italic><jats:sub>0<\/jats:sub>|<jats:sup><jats:italic>p<\/jats:italic><\/jats:sup>and show their performance in a small numerical study.<\/jats:p>","DOI":"10.1239\/jap\/1389370094","type":"journal-article","created":{"date-parts":[[2014,1,10]],"date-time":"2014-01-10T16:09:25Z","timestamp":1389370165000},"page":"969-982","source":"Crossref","is-referenced-by-count":11,"title":["Fractional Moments of Solutions to Stochastic Recurrence Equations"],"prefix":"10.1017","volume":"50","author":[{"given":"Thomas","family":"Mikosch","sequence":"first","affiliation":[]},{"given":"Gennady","family":"Samorodnitsky","sequence":"additional","affiliation":[]},{"given":"Laleh","family":"Tafakori","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2018,1,30]]},"reference":[{"key":"S0021900200120030_ref31","volume-title":"Probability Metrics and the Stability of Stochastic Models","author":"Rachev","year":"1991"},{"key":"S0021900200120030_ref26","doi-asserted-by":"publisher","DOI":"10.1137\/1129104"},{"key":"S0021900200120030_ref24","doi-asserted-by":"publisher","DOI":"10.3150\/12-BEJSP01"},{"key":"S0021900200120030_ref23","doi-asserted-by":"crossref","first-page":"90","DOI":"10.1214\/aoap\/1075828048","article-title":"AIMD algorithms and exponential functionals","volume":"14","author":"Guillemin","year":"2004","journal-title":"Ann. 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