{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,25]],"date-time":"2025-02-25T13:48:34Z","timestamp":1740491314394,"version":"3.38.0"},"reference-count":98,"publisher":"Cambridge University Press (CUP)","issue":"A","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. Appl. Probab."],"published-print":{"date-parts":[[2014,12]]},"abstract":"<jats:p>We survey some aspects of the classical prediction theory for stationary processes, in discrete time in Section 1, turning in Section 2 to continuous time, with particular reference to reproducing-kernel Hilbert spaces and the sampling theorem. We discuss the discrete-continuous theories of ARMA-CARMA, GARCH-COGARCH, and OPUC-COPUC in Section 3. We compare the various models treated in Section 4 by how well they model volatility, in particular volatility clustering. We discuss the infinite-dimensional case in Section 5, and turn briefly to applications in Section 6.<\/jats:p>","DOI":"10.1017\/s0021900200021276","type":"journal-article","created":{"date-parts":[[2016,3,30]],"date-time":"2016-03-30T16:25:31Z","timestamp":1459355131000},"page":"189-201","source":"Crossref","is-referenced-by-count":0,"title":["Aspects of prediction"],"prefix":"10.1017","volume":"51","author":[{"given":"N. H.","family":"Bingham","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Badr","family":"Missaoui","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2016,3,30]]},"reference":[{"volume-title":"Symmetric Multivariate and Related Distributions","year":"1990","key":"S0021900200021276_ref32"},{"doi-asserted-by":"publisher","key":"S0021900200021276_ref96","DOI":"10.1017\/S0305004100024944"},{"volume-title":"Gaussian Processes, Function Theory, and the Inverse Spectral Problem","year":"1976","key":"S0021900200021276_ref31"},{"volume-title":"Probability Distributions on Banach Spaces","year":"1987","key":"S0021900200021276_ref95"},{"doi-asserted-by":"publisher","key":"S0021900200021276_ref30","DOI":"10.1214\/aoms\/1177696684"},{"volume-title":"Orthogonal Polynomials","year":"1939","key":"S0021900200021276_ref94"},{"volume-title":"Szeg Ho's Theorem and its Descendants. Spectral Theory for L2 Perturbations of Orthogonal Polynomials","year":"2011","key":"S0021900200021276_ref93"},{"volume-title":"Orthogonal Polynomials on the Unit Circle. Part 2. Spectral Theory","year":"2005","key":"S0021900200021276_ref92"},{"volume-title":"Orthogonal Polynomials on the Unit Circle. Part 1. Classical Theory","year":"2005","key":"S0021900200021276_ref91"},{"doi-asserted-by":"publisher","key":"S0021900200021276_ref90","DOI":"10.1214\/aop\/1176991251"},{"key":"S0021900200021276_ref29","doi-asserted-by":"crossref","first-page":"299","DOI":"10.1215\/ijm\/1256053188","volume":"14","year":"1970","journal-title":"Illinois J. Math."},{"volume-title":"Stochastic Processes","year":"1953","key":"S0021900200021276_ref28"},{"key":"S0021900200021276_ref27","first-page":"54517","volume":"2006","year":"2006","journal-title":"IMRS Int. Math. Res. Surv."},{"volume-title":"Hilbert Spaces of Entire Functions","year":"1968","key":"S0021900200021276_ref26"},{"volume-title":"Radon Measures on Arbitrary Topological Spaces and Cylindrical Measures","year":"1973","key":"S0021900200021276_ref89"},{"volume-title":"Stationary and Related Stochastic Processes","year":"1967","key":"S0021900200021276_ref24"},{"volume-title":"Probability Theory on Vector Spaces","year":"1978","first-page":"237","key":"S0021900200021276_ref88"},{"key":"S0021900200021276_ref23","first-page":"7","volume":"28B","year":"1942","journal-title":"Ark. Mat. Astr. Fys."},{"volume-title":"Monte Carlo Statistical Methods","year":"2004","key":"S0021900200021276_ref87"},{"volume-title":"Prediction Theory and Harmonic Analysis: The Pesi Masani Volume","year":"1983","first-page":"37","key":"S0021900200021276_ref22"},{"doi-asserted-by":"publisher","key":"S0021900200021276_ref86","DOI":"10.4064\/sm207-3-2"},{"doi-asserted-by":"publisher","key":"S0021900200021276_ref21","DOI":"10.1080\/00036811003627567"},{"volume-title":"S\u00e9minaire de Probabilit\u00e9s XLIII","year":"2011","first-page":"191","key":"S0021900200021276_ref85"},{"doi-asserted-by":"publisher","key":"S0021900200021276_ref20","DOI":"10.1214\/105051606000000150"},{"volume-title":"Probability in Banach Spaces","year":"1992","first-page":"189","key":"S0021900200021276_ref84"},{"volume-title":"Time Series in the Time Domain","year":"1985","first-page":"279","key":"S0021900200021276_ref83"},{"doi-asserted-by":"publisher","key":"S0021900200021276_ref43","DOI":"10.1090\/S0273-0979-1985-15293-0"},{"volume-title":"Continuous time Econometric Modelling","year":"1990","key":"S0021900200021276_ref9"},{"doi-asserted-by":"publisher","key":"S0021900200021276_ref8","DOI":"10.1137\/1104040"},{"doi-asserted-by":"publisher","key":"S0021900200021276_ref7","DOI":"10.1090\/S0002-9947-1950-0051437-7"},{"doi-asserted-by":"publisher","key":"S0021900200021276_ref6","DOI":"10.1112\/plms\/pdq004"},{"doi-asserted-by":"publisher","key":"S0021900200021276_ref5","DOI":"10.1111\/j.1467-9868.2006.00569.x"},{"doi-asserted-by":"publisher","key":"S0021900200021276_ref4","DOI":"10.1016\/S0047-259X(03)00028-9"},{"doi-asserted-by":"publisher","key":"S0021900200021276_ref3","DOI":"10.1016\/j.laa.2009.04.023"},{"volume-title":"Probability Approximations via the Poisson Clumping Heuristic","year":"1989","key":"S0021900200021276_ref2"},{"volume":"233","volume-title":"Graduate Texts Math.","year":"2006","key":"S0021900200021276_ref1"},{"volume-title":"Applied Functional Data Analysis: Methods and Case Studies","year":"2002","key":"S0021900200021276_ref82"},{"volume-title":"Functional Data Analysis","year":"1997","key":"S0021900200021276_ref81"},{"key":"S0021900200021276_ref80","first-page":"323","volume":"3","year":"1967","journal-title":"Ann. Inst. H. Poincar\u00e9 B (N.S.)"},{"volume-title":"Time Series: Theory and Methods","year":"1991","key":"S0021900200021276_ref19"},{"doi-asserted-by":"publisher","key":"S0021900200021276_ref18","DOI":"10.1023\/A:1017972605872"},{"volume-title":"Stochastic Processes: Theory and Methods","year":"2001","first-page":"249","key":"S0021900200021276_ref17"},{"volume-title":"Time Series Analysis. Forecasting and Control","year":"2008","key":"S0021900200021276_ref16"},{"volume":"149","volume-title":"Lecture Notes Statist.","year":"2000","key":"S0021900200021276_ref15"},{"volume-title":"Interpolation, Identification, and Sampling","year":"1997","key":"S0021900200021276_ref79"},{"doi-asserted-by":"publisher","key":"S0021900200021276_ref14","DOI":"10.2748\/tmj\/1178241524"},{"volume-title":"Fourier Transforms in the Complex Domain","year":"1934","key":"S0021900200021276_ref78"},{"doi-asserted-by":"publisher","key":"S0021900200021276_ref13","DOI":"10.1111\/j.1467-9574.2010.00465.x"},{"volume":"1","volume-title":"Operators, Functions and Systems: An Easy Reading","year":"2002","key":"S0021900200021276_ref77"},{"doi-asserted-by":"publisher","key":"S0021900200021276_ref12","DOI":"10.1080\/03610926.2012.754467"},{"volume":"273","volume-title":"Grundl. Math. Wiss.","year":"1986","key":"S0021900200021276_ref76"},{"doi-asserted-by":"publisher","key":"S0021900200021276_ref11","DOI":"10.1214\/12-PS200"},{"key":"S0021900200021276_ref75","first-page":"47","volume":"591","year":"1975","journal-title":"Ann. Acad. Sci. Fenn. Ser. A I"},{"doi-asserted-by":"publisher","key":"S0021900200021276_ref10","DOI":"10.1214\/11-PS178"},{"doi-asserted-by":"publisher","key":"S0021900200021276_ref74","DOI":"10.1017\/S0266466600005296"},{"volume-title":"Symposium in Honour of Ole E. Barndorff-Nielsen","year":"2000","first-page":"61","key":"S0021900200021276_ref73"},{"doi-asserted-by":"publisher","key":"S0021900200021276_ref72","DOI":"10.1016\/0001-8708(68)90018-2"},{"doi-asserted-by":"publisher","key":"S0021900200021276_ref71","DOI":"10.1007\/s11785-009-0017-1"},{"volume-title":"Portfolio Selection: Efficient Diversification of Investments","year":"1959","key":"S0021900200021276_ref70"},{"key":"S0021900200021276_ref69","first-page":"77","volume":"7","year":"1952","journal-title":"J. Finance"},{"volume-title":"Prediction Theory and Harmonic Analysis: The Pesi Masani Volume","year":"1983","key":"S0021900200021276_ref68"},{"key":"S0021900200021276_ref67","first-page":"545","volume":"20","year":"1972","journal-title":"Indiana Univ. Math. J."},{"volume":"46","volume-title":"Graduate Texts Math.","year":"1978","key":"S0021900200021276_ref66"},{"volume-title":"Processus Stochastiques et Mouvement Brownien","year":"1948","first-page":"228","key":"S0021900200021276_ref65"},{"key":"S0021900200021276_ref64","first-page":"1","volume":"92","year":"1959","journal-title":"Trans. Amer. Math. Soc."},{"volume-title":"Processus Stochastiques et Mouvement Brownien","year":"1948","key":"S0021900200021276_ref63"},{"doi-asserted-by":"publisher","key":"S0021900200021276_ref62","DOI":"10.1007\/BF02391767"},{"doi-asserted-by":"publisher","key":"S0021900200021276_ref61","DOI":"10.1002\/sapm195938168"},{"volume":"2","volume-title":"Selected Works of A. N. Kolmogorov","year":"1986","key":"S0021900200021276_ref60"},{"key":"S0021900200021276_ref59","first-page":"1","volume":"2","year":"1941","journal-title":"Bull. Moskov. Gos. Univ. Mat."},{"doi-asserted-by":"publisher","key":"S0021900200021276_ref58","DOI":"10.1007\/978-3-540-30788-4_21"},{"doi-asserted-by":"publisher","key":"S0021900200021276_ref57","DOI":"10.1239\/jap\/1091543413"},{"volume-title":"An Introduction to Harmonic Analysis","year":"2004","key":"S0021900200021276_ref56"},{"key":"S0021900200021276_ref55","first-page":"1363","volume":"366","year":"2014","journal-title":"Trans. Amer. Math. Soc."},{"doi-asserted-by":"publisher","key":"S0021900200021276_ref54","DOI":"10.1007\/BF02590624"},{"volume-title":"Foundations of Modern Probability","year":"2002","key":"S0021900200021276_ref53"},{"volume-title":"Stochastic Processes and Functional Analysis: In Celebration of M. M. Rao's 65th Birthday","year":"1997","first-page":"123","key":"S0021900200021276_ref52"},{"volume-title":"Multidimensional Second Order Stochastic Processes","year":"1997","key":"S0021900200021276_ref51"},{"volume-title":"From Stochastic Calculus to Mathematical Finance: The Shiryaev Festschrift","year":"2006","key":"S0021900200021276_ref50"},{"volume":"129","volume-title":"Camb. Tracts Math.","year":"1997","key":"S0021900200021276_ref49"},{"key":"S0021900200021276_ref48","doi-asserted-by":"crossref","first-page":"209","DOI":"10.1215\/kjm\/1250777359","volume":"28","year":"1954","journal-title":"Mem. Coll. Sci. Univ. Kyoto Ser. A. Math."},{"doi-asserted-by":"publisher","key":"S0021900200021276_ref47","DOI":"10.1214\/009053606000000209"},{"key":"S0021900200021276_ref46","doi-asserted-by":"crossref","first-page":"523","DOI":"10.1007\/s00440-007-0074-1","volume":"140","year":"2008","journal-title":"Prob. Theory Relat. Fields"},{"volume-title":"Sampling Theory in Fourier and Signal Analysis: Advanced Topics","year":"1996","key":"S0021900200021276_ref45"},{"volume-title":"Sampling Theory in Fourier and Signal Analysis: Foundations","year":"1996","key":"S0021900200021276_ref44"},{"doi-asserted-by":"publisher","key":"S0021900200021276_ref41","DOI":"10.1111\/j.1368-423X.2007.00210.x"},{"key":"S0021900200021276_ref40","doi-asserted-by":"crossref","first-page":"31","DOI":"10.4064\/sm-91-1-31-43","volume":"91","year":"1988","journal-title":"Studia Math."},{"volume-title":"Toeplitz Forms and Their Applications","year":"1958","key":"S0021900200021276_ref39"},{"volume-title":"ARCH Models and Financial Applications","year":"1997","key":"S0021900200021276_ref38"},{"key":"S0021900200021276_ref37","first-page":"235","volume":"69","year":"1980","journal-title":"Studia Math."},{"doi-asserted-by":"publisher","key":"S0021900200021276_ref36","DOI":"10.1007\/s10440-008-9414-0"},{"volume":"4","volume-title":"Generalized Functions","year":"1964","key":"S0021900200021276_ref35"},{"volume":"III","volume-title":"Collected Works With Commentaries: The Hopf-Wiener Integral Equation; Prediction and Filtering; Quantum Mechanics and Relativity; Miscellaneous Mathematical Papers","year":"1981","key":"S0021900200021276_ref99"},{"volume-title":"Extreme Values in Finance, Telecommunications, and the Environment","year":"2004","key":"S0021900200021276_ref34"},{"volume-title":"Extrapolation, Interpolation and Smoothing of Stationary Time Series. With Engineering Applications","year":"1949","key":"S0021900200021276_ref98"},{"volume-title":"The Oxford Handbook of Functional Data Analysis","year":"2010","key":"S0021900200021276_ref33"},{"volume":"33","volume-title":"Camb. Tracts Math.","year":"1935","key":"S0021900200021276_ref97"},{"volume":"72","volume-title":"Camb. Tracts Math.","year":"1977","key":"S0021900200021276_ref42"}],"container-title":["Journal of Applied Probability"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0021900200021276","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,6,16]],"date-time":"2022-06-16T13:42:03Z","timestamp":1655386923000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0021900200021276\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,12]]},"references-count":98,"journal-issue":{"issue":"A","published-print":{"date-parts":[[2014,12]]}},"alternative-id":["S0021900200021276"],"URL":"https:\/\/doi.org\/10.1017\/s0021900200021276","relation":{},"ISSN":["0021-9002","1475-6072"],"issn-type":[{"type":"print","value":"0021-9002"},{"type":"electronic","value":"1475-6072"}],"subject":[],"published":{"date-parts":[[2014,12]]}}}