Journal of Applied Probability

Aspects of prediction

N. H. Bingham and Badr Missaoui

Abstract

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.

Article information

Source
J. Appl. Probab., Volume 51A (2014), 189-201.

Dates
First available in Project Euclid: 2 December 2014

Permanent link to this document
https://projecteuclid.org/euclid.jap/1417528475

Digital Object Identifier
doi:10.1239/jap/1417528475

Mathematical Reviews number (MathSciNet)
MR3317358

Zentralblatt MATH identifier
1314.62214

Subjects
Primary: 60-02: Research exposition (monographs, survey articles)
Secondary: 62-02: Research exposition (monographs, survey articles)

Keywords
Stationary process Kolmogorov isomorphism theorem time series stochastic volatility volatility clustering Banach space covariance operator locally convex reproducing-kernel Hilbert space sampling theorem

Citation

Bingham, N. H.; Missaoui, Badr. Aspects of prediction. J. Appl. Probab. 51A (2014), 189--201. doi:10.1239/jap/1417528475. https://projecteuclid.org/euclid.jap/1417528475


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References

  • Albiac, F. and Kalton, N. J. (2006). Topics in Banach Space Theory (Graduate Texts Math. 233). Springer, New York.
  • Aldous, D. (1989). Probability Approximations via the Poisson Clumping Heuristic. Springer, New York.
  • Alpay, D., Timoshenko, O. and Volok, D. (2009). Carathéodory–Fejér interpolation in locally convex topological vector spaces. Linear Algebra Appl. 431, 1257–1266.
  • Antoniadis, A. and Sapatinas, T. (2003). Wavelet methods for continuous-time prediction using Hilbert-valued autoregressive processes. J. Multivariate Anal. 87, 133–158.
  • Antoniadis, A., Paparoditis, E. and Sapatinas, T. (2006). A functional wavelet-kernel approach for time series prediction. J. R. Statist. Soc. B 68, 837–857.
  • Applebaum, D. and Riedle, M. (2010). Cylindrical Lévy processes in Banach spaces. Proc. London Math. Soc. 101, 697–726.
  • Aronszajn, N. (1950). Theory of reproducing kernels. Trans. Amer. Math. Soc. 68, 337–404.
  • Belyaev, Y. K. (1959). Analytic random processes. Theory Prob. Appl. 4, 402–409.
  • Bergstrom, A. R. (1990). Continuous time Econometric Modelling. Oxford University Press.
  • Bingham, N. H. (2012). Szegö's theorem and its probabilistic descendants. Prob. Surveys 9, 287–324.
  • Bingham, N. H. (2012). Multivariate prediction and matrix Szegö theory. Prob. Surveys 9, 325–339.
  • Bingham, N. H. (2014). Modelling and prediction of financial time series. Commun. Statist. Theory Meth. 43, 1351–1361.
  • Bingham, N. H., Fry, J. M. and Kiesel, R. (2010). Multivariate elliptic processes. Statist. Neerlandica 64, 352–366.
  • Boas, R. P., Jr. (1972). Summation formulas and band-limited signals. Tôhoku Math. J. 24, 121–125.
  • Bosq, D. (2000). Linear Processes in Function Spaces. Theory and Applications (Lecture Notes Statist. 149). Springer, New York.
  • Box, G. E. P., Jenkins, G. M. and Reinsel, G. C. (2008). Time Series Analysis. Forecasting and Control, 4th edn. John Wiley, Hoboken, NJ.
  • Brockwell, P. J. (2001). Continuous-time ARMA processes. In Stochastic Processes: Theory and Methods (Handbook Statist. 19), Elsevier, London, pp. 249–276.
  • Brockwell, P. J. (2001). Lévy-driven CARMA processes. Ann. Inst. Statist. Math. 53, 113–124.
  • Brockwell, P. J. and Davis, R. A. (1991). Time Series: Theory and Methods, 2nd edn. Springer, New York.
  • Brockwell, P. J., Chadraa, E. and Lindner, A. (2006). Continuous-time GARCH processes. Ann. Appl. Prob. 16, 790–826.
  • Butzer, P. L. et al. (2011). The sampling theorem, Poisson's summation formula, general Parseval formula, reproducing kernel formula and the Paley–Wiener theorem for bandlimited signals–-their interconnections. Appl. Anal. 90, 431–461.
  • Chobanyan, S. A. and Vakhania, N. N. (1983). The linear prediction and approximation of weak second order random elements. In Prediction Theory and Harmonic Analysis: The Pesi Masani Volume, eds V. Mandrekar and H. Saleti, North-Holland, Amsterdam, pp. 37–60.
  • Cramér, H. (1942). On harmonic analysis in certain function spaces. Ark. Mat. Astr. Fys. 28B, 7pp.
  • Cramér, H. and Leadbetter, M. R. (1967). Stationary and Related Stochastic Processes. John Wiley, New York.
  • Cugliari, J. (2011). Prévision non paramétrique de processus à valeurs fonctionnelles. Application à la consommation d'électricité. Doctoral Thesis, Université Paris-Sud XI.
  • De Branges, L. (1968). Hilbert Spaces of Entire Functions. Prentice-Hall, Englewood Cliffs, NJ.
  • Denisov, S. A. (2006). Continuous analogs of polynomials orthogonal on the unit circle and Krein systems. IMRS Int. Math. Res. Surv. 2006, 54517.
  • Doob, J. L. (1953). Stochastic Processes. John Wiley, New York.
  • Dym, H. and McKean, H. P., Jr. (1970). Application of de Branges spaces of integral functions to prediction of stationary Gaussian processes. Illinois J. Math. 14, 299–343.
  • Dym, H. and McKean, H. P. (1970). Extrapolation and interpolation of stationary Gaussian processes. Ann. Math. Statist. 41, 1817–1844.
  • Dym, H. and McKean, H. P. (1976). Gaussian Processes, Function Theory, and the Inverse Spectral Problem. Academic Press, New York.
  • Fang, K. T., Kotz, S. and Ng, K. W. (1990). Symmetric Multivariate and Related Distributions. Chapman and Hall, London.
  • Ferraty, F. and Romain, Y. (eds) (2010). The Oxford Handbook of Functional Data Analysis. Oxford University Press.
  • Finkenstädt, B. and Rootzén, H. (eds) (2004). Extreme Values in Finance, Telecommunications, and the Environment. Chapman & Hall/CRC, Boca Raton, FL.
  • Gel'fand, I. M., and Vilenkin, N. Y. (1964). Generalized Functions, Vol. 4. Academic Press, New York.
  • Ginovyan, M. S. and Mikaelyan, L. V. (2010). Prediction error for continuous-time stationary processes with singular spectral densities. Acta Appl. Math. 110, 327–351.
  • Gorniak, J. and Weron, A. (1980/81). Aronszajn–Kolmogorov type theorems for positive definite kernels in locally convex spaces. Studia Math. 69, 235–246.
  • Gouriéroux, C. (1997). ARCH Models and Financial Applications. Springer, New York.
  • Grenander, U. and Szegö, G. (1958). Toeplitz Forms and Their Applications. University of California Press, Berkeley, CA.
  • Hajduk-Chmielewska, G. (1988). The Wold-Cramér concordance problem for Banach-space-valued stationary processes. Studia Math. 91, 31–43.
  • Haug, S., Klüppelberg, C., Lindner, A. and Zapp, M. (2007). Method of moment estimation in the COGARCH(1,1) model. Econometric J. 10, 320–341.
  • Higgins, J. R. (1977). Completeness and Basis Properties of Sets of Special Functions (Camb. Tracts Math. 72). Cambridge University Press.
  • Higgins, J. R. (1985). Five short stories about the cardinal series. Bull. Amer. Math. Soc. (N.S.) 12, 45–89.
  • Higgins, J. R. (1996). Sampling Theory in Fourier and Signal Analysis: Foundations. Clarendon Press, Oxford.
  • Higgins, J. R. and Stens, R. L. (eds) (1996). Sampling Theory in Fourier and Signal Analysis: Advanced Topics. Oxford University Press.
  • Inoue, A. (2008). AR and MA representations of partial autocorrelation functions, with applications. Prob. Theory Relat. Fields 140, 523–551.
  • Inoue, A. and Kasahara, Y. (2006). Explicit representation of finite predictor coefficients and its applications. Ann. Statist. 34, 973–993.
  • Itô, K. (1954). Stationary random distributions. Mem. Coll. Sci. Univ. Kyoto Ser. A. Math. 28, 209–223.
  • Janson, S. (1997). Gaussian Hilbert Spaces (Camb. Tracts Math. 129). Cambridge University Press.
  • Kabanov, Y., Liptser, R. and Stoyanov, J. (eds) (2006). From Stochastic Calculus to Mathematical Finance: The Shiryaev Festschrift. Springer, Berlin.
  • Kakihara, Y. (1997). Multidimensional Second Order Stochastic Processes. World Scientific, River Edge, NJ.
  • Kakihara, Y. (1997). Dilations of Hilbert–Schmidt operator-valued measures and applications. In Stochastic Processes and Functional Analysis: In Celebration of M. M. Rao's 65th Birthday, eds J. A. Goldstein, N. E. Gretsky and J. J. Uhl, Dekker, New York, pp. 123–135.
  • Kallenberg, O. (2002). Foundations of Modern Probability, 2nd edn. Springer, New York.
  • Karhunen, K. (1950). Uber die Struktur stationärer zufälliger funktionen. Ark. Mat. 1, 141–160.
  • Kasahara, Y. and Bingham, N. H. (2014). Verblunsky coefficients and Nehari sequences. Trans. Amer. Math. Soc. 366, 1363–1378.
  • Katznelson, Y. (2004). An Introduction to Harmonic Analysis, 3rd edn. Cambridge University Press.
  • Klüppelberg, C., Lindner, A. and Maller, R. (2004). A continuous-time GARCH process driven by a Lévy process: stationarity and second-order behaviour. J. Appl. Prob. 41, 601–622.
  • Klüppelberg, C., Lindner, A. and Maller, R. A. (2006). Continuous-time volatility modelling: COGARCH versus Ornstein-Uhlenbeck models. In From Stochastic Calculus to Mathematical Finance: The Shiryaev Festschrift, eds Y. Kabanov et al., Springer, Berlin, pp. 393–419.
  • Kolmogorov, A. N. (1941). Stationary sequences in Hilbert space. Bull. Moskov. Gos. Univ. Mat. 2, 1–40 (in Russian).
  • Kolmogorov, A. N. (1986). Selected Works of A. N. Kolmogorov, Vol. 2. Nauka, Moskva (in Russian).
  • Kramer, H. P. (1959). A generalised sampling theorem. J. Math. Phys. 38, 68–72.
  • Levinson, N. and McKean, H. P., Jr. (1964). Weighted trigonometrical approximation on $R^1$ with application to the germ field of a stationary Gaussian noise. Acta Math. 112, 99–143.
  • Lévy, P. (1948). Processus Stochastiques et Mouvement Brownien. Gauthier-Villars, Paris.
  • Lloyd, S. P. (1959). A sampling theorem for stationary (wide sense) stochastic processes. Trans. Amer. Math. Soc. 92, 1–12.
  • Loève, M. (1948). Fonctions aléatoires du second ordre. Supplement to Processus Stochastiques et Mouvement Brownien, Gauthier-Villars, Paris, pp. 228–352.
  • Loève, M. (1978). Probability Theory. II (Graduate Texts Math. 46), 4th edn. Springer, New York.
  • Mandrekar, V. and Salehi, H. (1972). The square-integrability of operator-valued functions with respect to a non-negative operator-valued measure and the Kolmogorov isomorphism theorem. Indiana Univ. Math. J. 20, 545–563.
  • Mandrekar, V. and Salehi, H. (eds) (1983). Prediction Theory and Harmonic Analysis: The Pesi Masani Volume. North-Holland, Amsterdam.
  • Markowitz, H. (1952). Portfolio selection. J. Finance 7, 77–91.
  • Markowitz, H. M. (1959). Portfolio Selection: Efficient Diversification of Investments. John Wiley, New York.
  • Martin, R. T. W. (2010). Symmetric operators and reproducing kernel Hilbert spaces. Complex Anal. Operat. Theory 4, 845–880.
  • Masani, P. (1968). Orthogonally scattered measures. Adv. Math. 2, 61–117.
  • Nelson, D. B. (1990). Stationarity and persistence in the GARCH(1,1) model. Econometric Theory 6, 318–334.
  • Niemi, H. (1975). Stochastic processes as Fourier transforms of stochastic measures. Ann. Acad. Sci. Fenn. Ser. A I 591, 47pp.
  • Nikol'skiǐ, N. K. (1986). Treatise on the Shift Operator. Spectral Function Theory (Grundl. Math. Wiss. 273). Springer, Berlin.
  • Nikolski, N. K. (2002). Operators, Functions and Systems: An Easy Reading, Vol. 1. American Mathematical Society, Providence, RI.
  • Paley, R. E. A. C. and Wiener, N. (1934). Fourier Transforms in the Complex Domain. American Mathematical Society, New York.
  • Partington, J. R. (1997). Interpolation, Identification, and Sampling. Oxford University Press.
  • Payen, R. (1967). Fonctions aléatoires du second ordre à valeurs dans un espace de Hilbert. Ann. Inst. H. Poincaré B (N.S.) 3, 323–396.
  • Ramsay, J. O. and Silverman, B. W. (1997). Functional Data Analysis. Springer, New York.
  • Ramsay, J. O. and Silverman, B. W. (2002). Applied Functional Data Analysis: Methods and Case Studies. Springer, New York.
  • Rao, M. M. (1985). Harmonizable, Cramér, and Karhunen classes of processes. In Time Series in the Time Domain (Handbook Statist. 5), North-Holland, Amsterdam, pp. 279–310.
  • Richard, P. H. (1992). Harmonizability, $V$-boundedness and stationary dilation for Banach-valued processes. In Probability in Banach Spaces, eds R. M. Dudley, M. G. Hahn and J. Kuelbs, Birkhäuser, Basel, pp. 189–205.
  • Riedle, M. (2011). Cylindrical Wiener processes. In Séminaire de Probabilités XLIII (Lecture Notes Math. 2006), Springer, Berlin, pp. 191–214.
  • Riedle, M. (2011). Infinitely divisible cylindricalmeasures on Banach spaces. Studia Math. 207, 235–256.
  • Robert, C. P. and Casella, G. (2004). Monte Carlo Statistical Methods. Springer, New York.
  • Schmidt, F. (1978). Banach-space-valued stationary processes with absolutely continuous spectral function. In Probability Theory on Vector Spaces (Lecture Notes Math. 656), Springer, Berlin, pp. 237–244.
  • Schwartz, L. (1973). Radon Measures on Arbitrary Topological Spaces and Cylindrical Measures. Oxford University Press.
  • Shiryaev, A. N. (1989). Kolmogorov: life and creative activities. Ann. Prob. 17, 866–944.
  • Simon, B. (2005). Orthogonal Polynomials on the Unit Circle. Part 1. Classical Theory. American Mathematical Society, Providence, RI.
  • Simon, B. (2005). Orthogonal Polynomials on the Unit Circle. Part 2. Spectral Theory. American Mathematical Society, Providence, RI.
  • Simon, B. (2011). Szegő's Theorem and its Descendants. Spectral Theory for $L^2$ Perturbations of Orthogonal Polynomials. Princeton University Press.
  • Szegö, G. (1939). Orthogonal Polynomials. American Mathematical Society, New York.
  • Vakhania, N. N., Tarieladze, V. I. and Chobanyan, S. A. (1987). Probability Distributions on Banach Spaces. Reidel, Dordrecht.
  • Weston, J. D. (1949). The cardinal series in Hilbert space. Proc. Camb. Phil. Soc. 45, 335–341.
  • Whittaker, J. M. (1935). Interpolatory Function Theory (Camb. Tracts Math. 33). Cambridge University Press.
  • Wiener, N. (1949). Extrapolation, Interpolation and Smoothing of Stationary Time Series. With Engineering Applications. John Wiley, New York.
  • Wiener, N. (1981). Collected Works With Commentaries: The Hopf–Wiener Integral Equation; Prediction and Filtering; Quantum Mechanics and Relativity; Miscellaneous Mathematical Papers (ed. P. R. Masani), Vol. III. MIT Press, Cambridge, MA.