{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,3]],"date-time":"2026-04-03T08:37:15Z","timestamp":1775205435832,"version":"3.50.1"},"reference-count":7,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2011,11,25]],"date-time":"2011-11-25T00:00:00Z","timestamp":1322179200000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Econom. Theory"],"published-print":{"date-parts":[[2012,6]]},"abstract":"<jats:p>We discuss the moment condition for the fractional functional central limit theorem (FCLT) for partial sums of <jats:italic>x<jats:sub>t<\/jats:sub><\/jats:italic> = \u0394<jats:sup>\u2212<jats:italic>d<\/jats:italic><\/jats:sup><jats:italic>u<jats:sub>t<\/jats:sub><\/jats:italic>, where <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0266466611000697_inline1\"><jats:alt-text>$d\\, \\in \\,\\left({ - {1 \\over 2}\\,,\\,{1 \\over 2}} \\right)$<\/jats:alt-text><\/jats:inline-graphic> is the fractional integration parameter and <jats:italic>u<jats:sub>t<\/jats:sub><\/jats:italic> is weakly dependent. The classical condition is existence of <jats:italic>q<\/jats:italic> \u2265 2 and <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0266466611000697_inline2\"><jats:alt-text>$q\\, &gt; \\,\\left( {d\\, + \\,{1 \\over 2}} \\right)^{ - 1} $<\/jats:alt-text><\/jats:inline-graphic> moments of the innovation sequence. When <jats:italic>d<\/jats:italic> is close to <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0266466611000697_inline3\"><jats:alt-text>$ - {1 \\over 2}$<\/jats:alt-text><\/jats:inline-graphic> this moment condition is very strong. Our main result is to show that when <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0266466611000697_inline4\"><jats:alt-text>$d\\, \\in \\,\\left({ - \\,{1 \\over 2},\\,0} \\right)$<\/jats:alt-text><\/jats:inline-graphic> and under some relatively weak conditions on <jats:italic>u<jats:sub>t<\/jats:sub><\/jats:italic>, the existence of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0266466611000697_inline5\"><jats:alt-text>$q\\, \\ge \\,\\left({d\\, + \\,{1 \\over 2}} \\right)^{ - 1} $<\/jats:alt-text><\/jats:inline-graphic> moments is in fact necessary for the FCLT for fractionally integrated processes and that <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0266466611000697_inline6\"><jats:alt-text>$q\\, &gt; \\,\\left( {d\\, + \\,{1 \\over 2}} \\right)^{ - 1} $<\/jats:alt-text><\/jats:inline-graphic> moments are necessary for more general fractional processes. Davidson and de Jong (2000, <jats:italic>Econometric Theory<\/jats:italic> 16, 643\u2013666) presented a fractional FCLT where only <jats:italic>q<\/jats:italic> &gt; 2 finite moments are assumed. As a corollary to our main theorem we show that their moment condition is not sufficient and hence that their result is incorrect.<\/jats:p>","DOI":"10.1017\/s0266466611000697","type":"journal-article","created":{"date-parts":[[2011,11,25]],"date-time":"2011-11-25T10:45:02Z","timestamp":1322217902000},"page":"671-679","source":"Crossref","is-referenced-by-count":24,"title":["A NECESSARY MOMENT CONDITION FOR THE FRACTIONAL FUNCTIONAL CENTRAL LIMIT THEOREM"],"prefix":"10.1017","volume":"28","author":[{"given":"S\u00f8ren","family":"Johansen","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Morten","family":"\u00d8rregaard Nielsen","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2011,11,25]]},"reference":[{"key":"S0266466611000697_ref5","doi-asserted-by":"publisher","DOI":"10.1016\/S0304-4149(99)00088-5"},{"key":"S0266466611000697_ref4","doi-asserted-by":"publisher","DOI":"10.1137\/1115050"},{"key":"S0266466611000697_ref1","volume-title":"Convergence of Probability Measures","author":"Billingsley","year":"1968"},{"key":"S0266466611000697_ref2","volume-title":"Regular Variation","author":"Bingham","year":"1989"},{"key":"S0266466611000697_ref3","doi-asserted-by":"publisher","DOI":"10.1017\/S0266466600165028"},{"key":"S0266466611000697_ref6","doi-asserted-by":"publisher","DOI":"10.1007\/BF00532868"},{"key":"S0266466611000697_ref7","doi-asserted-by":"publisher","DOI":"10.1214\/074921706000000572"}],"container-title":["Econometric Theory"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0266466611000697","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,4,25]],"date-time":"2019-04-25T20:55:57Z","timestamp":1556225757000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0266466611000697\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,11,25]]},"references-count":7,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2012,6]]}},"alternative-id":["S0266466611000697"],"URL":"https:\/\/doi.org\/10.1017\/s0266466611000697","relation":{},"ISSN":["0266-4666","1469-4360"],"issn-type":[{"value":"0266-4666","type":"print"},{"value":"1469-4360","type":"electronic"}],"subject":[],"published":{"date-parts":[[2011,11,25]]}}}