{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,17]],"date-time":"2026-08-17T20:46:28Z","timestamp":1786999588382,"version":"build-2736575974"},"reference-count":18,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2010,10,18]],"date-time":"2010-10-18T00:00:00Z","timestamp":1287360000000},"content-version":"unspecified","delay-in-days":7991,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Econ Theory"],"published-print":{"date-parts":[[1988,12]]},"abstract":"<jats:p>\n                    Under general conditions the sample covariance matrix of a vector martingale and its differences converges weakly to the matrix stochastic integral \u222b\n                    <jats:sub>0<\/jats:sub>\n                    <jats:sup>1<\/jats:sup>\n                    <jats:italic>BdB\u2032<\/jats:italic>\n                    , where\n                    <jats:italic>B<\/jats:italic>\n                    is vector Brownian motion. For strictly stationary and ergodic sequences, rather than martingale differences, a similar result obtains. In this case, the limit is \u222b\n                    <jats:sub>0<\/jats:sub>\n                    <jats:sup>1<\/jats:sup>\n                    <jats:italic>BdB\u2032<\/jats:italic>\n                    + \u039b and involves a constant matrix \u039b of bias terms whose magnitude depends on the serial correlation properties of the sequence. This note gives a simple proof of the result using martingale approximations.\n                  <\/jats:p>","DOI":"10.1017\/s026646660001344x","type":"journal-article","created":{"date-parts":[[2010,10,18]],"date-time":"2010-10-18T05:12:04Z","timestamp":1287378724000},"page":"528-533","source":"Crossref","is-referenced-by-count":50,"title":["Weak Convergence of Sample Covariance Matrices to Stochastic Integrals Via Martingale Approximations"],"prefix":"10.1017","volume":"4","author":[{"given":"P.C.B.","family":"Phillips","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2010,10,18]]},"reference":[{"key":"S026646660001344X_ref012","unstructured":"Phillips, P.C.B. Partially identified econometric models. Cowles Foundation Discussion Paper No. 845, Yale University, July 1987."},{"key":"S026646660001344X_ref017","doi-asserted-by":"publisher","DOI":"10.2307\/1911260"},{"key":"S026646660001344X_ref003","volume-title":"Martingale limit theory and its application","author":"Hall","year":"1980"},{"key":"S026646660001344X_ref005","doi-asserted-by":"publisher","DOI":"10.1017\/S0266466600012287"},{"key":"S026646660001344X_ref016","doi-asserted-by":"publisher","DOI":"10.1080\/01621459.1984.10477111"},{"key":"S026646660001344X_ref002","doi-asserted-by":"publisher","DOI":"10.1214\/aos\/1176350711"},{"key":"S026646660001344X_ref014","doi-asserted-by":"publisher","DOI":"10.2307\/2297602"},{"key":"S026646660001344X_ref009","doi-asserted-by":"publisher","DOI":"10.1016\/0047-259X(88)90039-5"},{"key":"S026646660001344X_ref004","doi-asserted-by":"publisher","DOI":"10.1017\/S0266466600013402"},{"key":"S026646660001344X_ref007","doi-asserted-by":"publisher","DOI":"10.2307\/1913237"},{"key":"S026646660001344X_ref013","doi-asserted-by":"publisher","DOI":"10.1090\/conm\/080\/999009"},{"key":"S026646660001344X_ref008","doi-asserted-by":"publisher","DOI":"10.1017\/S0266466600004126"},{"key":"S026646660001344X_ref011","doi-asserted-by":"publisher","DOI":"10.2307\/1911357"},{"key":"S026646660001344X_ref010","doi-asserted-by":"publisher","DOI":"10.1093\/biomet\/74.3.535"},{"key":"S026646660001344X_ref015","unstructured":"Sims, C.A. , Stock, J.H. & Watson, M.W. . Inference in linear time-series models with some unit roots. Mimeographed, Minnesota University, 1986."},{"key":"S026646660001344X_ref018","unstructured":"Stock, J.H. & Watson, M.W. . Testing for common trends. Hoover Institution Working Paper No. E-87\u20132, 1987."},{"key":"S026646660001344X_ref001","volume-title":"Time series: data analysis and theory","author":"Brillinger","year":"1981"},{"key":"S026646660001344X_ref006","doi-asserted-by":"publisher","DOI":"10.1016\/0304-4076(86)90001-1"}],"container-title":["Econometric Theory"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S026646660001344X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,11,9]],"date-time":"2021-11-09T00:02:03Z","timestamp":1636416123000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S026646660001344X\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1988,12]]},"references-count":18,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1988,12]]}},"alternative-id":["S026646660001344X"],"URL":"https:\/\/doi.org\/10.1017\/s026646660001344x","relation":{},"ISSN":["0266-4666","1469-4360"],"issn-type":[{"value":"0266-4666","type":"print"},{"value":"1469-4360","type":"electronic"}],"subject":[],"published":{"date-parts":[[1988,12]]}}}