Application of the theory of KM2O-Langevin equations to the linear prediction problem for the multi-dimensional weakly stationary time series

Yasunori OKABE

doi:10.2969/jmsj/04520277

Application of the theory of KM₂O-Langevin equations to the linear prediction problem for the multi-dimensional weakly stationary time series

Yasunori OKABE

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JOURNAL FREE ACCESS

1993 Volume 45 Issue 2 Pages 277-294

DOI https://doi.org/10.2969/jmsj/04520277

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Published: 1993 Received: February 07, 1992 Available on J-STAGE: October 20, 2006 Accepted: - Advance online publication: - Revised: -
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Date of correction: October 20, 2006 Reason for correction: - Correction: SUBTITLE Details: Wrong : Dedicated to Professor Hiroshi Tanaka on his sixtieth birthday

Date of correction: October 20, 2006 Reason for correction: - Correction: CITATION Details: Wrong : 1) J. Durbin, The fitting of time series models, Rev. Int. Stat., 28 (1960), 233-244.
2) N. Levinson, The Wiener RMS error criterion in filter design and prediction, J. Math. and Phys., 25 (1947), 261-278.
3) P. Masani, The prediction theory of multivariate stochastic processes, III. Unbounded spectral densities, Acta Math., 104 (1960), 141-162.
4) P. Masani and N. Wiener, Non-linear prediction, Probability and Statistics, The Harald Cramér Volume (ed. U. Grenander), John Wiley, 1959, pp. 190-212.
5) T. Miyoshi, On (l, m)-string and (α, β, γ, δ)-Langevin equation associated with a stationary Gaussian process, J. Fac. Sci. Univ. Tokyo, Sect. IA Math., 30 (1983), 139-190.
6) T. Miyoshi, On an R^d-valued stationary Gaussian process associated with (k, l, m)-string and (α, β, γ, δ)-Langevin equation, J. Fac. Sci. Univ. Tokyo, Sect. IA Math., 31 (1984), 154-194.
7) Y. Okabe, On stochastic difference equations for the multi-dimensional weakly stationary time series, Prospect of Algebraic Analysis (eds. M. Kashiwara and T. Kawai), Academic Press, Tokyo, 1988, pp. 601-645.
8) Y. Okabe, The random forces associated with discrete-time weakly stationary processes, Proceedings of Preseminar for International Conference on Gaussian Random fields, Nagoya, 1991, pp. 126-134.
9) Y. Okabe, Langevin equation and causal analysis, Sugaku, 43 (1991), 322-346 (in Japanese).
10) Y. Okabe and A. Inoue, The theory of KM₂O-Langevin equations and its applications to data analysis (II): Causal analysis, to be submitted in Nagoya Math. J..
11) Y. Okabe and Y. Nakano, The theory of KM₂O-Langevin equations and its applications to data analysis (I): Stationary analysis, Hokkaido Math. J., 20 (1991), 45-90.
12) Y. Okabe and T. Ootsuka, Application of the theory of KM₂O-Langevin equations to the non-linear prediction problem for the one-dimensional strictly stationary time series, to be submitted in J. Math. Soc. Japan.
13) Y. Okabe and T. Ootsuka, The theory of KM₂O-Langevin equations and its applications to data analysis (III): Prediction analysis, in preparation.
14) J. von Neumann, Functional Operators, Vol. 2, Princeton, 1950.
15) P. Whittle, On the fitting of multivariate autoregressions, and the approximate canonical factorization of a spectral density matrix, Biomerika, 50 (1963), 129-134.
16) N. Wiener and P. Masani, The prediction theory of multivariate stochastic processes, I. The regularity condition, Acta Math., 98 (1957), 111-150.
17) N. Wiener and P. Masani, The prediction theory of multivariate stochastic processes, II. The linear predictor, Acta Math., 99 (1958), 93-137.
18) R. A. Wiggins and E. A. Robinson, Recursive solution to the multichannel fitting problem, J. Geophys. Res., 70 (1965), 1885-1891.

Right : [1] J. Durbin, The fitting of time series models, Rev. Int. Stat., 28 (1960), 233-244.
[2] N. Levinson, The Wiener RMS error criterion in filter design and prediction, J. Math. and Phys., 25 (1947), 261-278.
[3] P. Masani, The prediction theory of multivariate stochastic processes, III. Unbounded spectral densities, Acta Math., 104 (1960), 141-162.
[4] P. Masani and N. Wiener, Non-linear prediction, Probability and Statistics, The Harald Cramér Volume (ed. U. Grenander), John Wiley, 1959, pp. 190-212.
[5] T. Miyoshi, On (l, m)-string and (α, β, γ, δ)-Langevin equation associated with a stationary Gaussian process, J. Fac. Sci. Univ. Tokyo, Sect. IA Math., 30 (1983), 139-190.
[6] T. Miyoshi, On an R^d-valued stationary Gaussian process associated with (k, l, m)-string and (α, β, γ, δ)-Langevin equation, J. Fac. Sci. Univ. Tokyo, Sect. IA Math., 31 (1984), 154-194.
[7] Y. Okabe, On stochastic difference equations for the multi-dimensional weakly stationary time series, Prospect of Algebraic Analysis (eds. M. Kashiwara and T. Kawai), Academic Press, Tokyo, 1988, pp. 601-645.
[8] Y. Okabe, The random forces associated with discrete-time weakly stationary processes, Proceedings of Preseminar for International Conference on Gaussian Random fields, Nagoya, 1991, pp. 126-134.
[9] Y. Okabe, Langevin equation and causal analysis, Sugaku, 43 (1991), 322-346 (in Japanese).
[10] Y. Okabe and A. Inoue, The theory of KM₂O-Langevin equations and its applications to data analysis (II): Causal analysis, to be submitted in Nagoya Math. J..
[11] Y. Okabe and Y. Nakano, The theory of KM₂O-Langevin equations and its applications to data analysis (I): Stationary analysis, Hokkaido Math. J., 20 (1991), 45-90.
[12] Y. Okabe and T. Ootsuka, Application of the theory of KM₂O-Langevin equations to the non-linear prediction problem for the one-dimensional strictly stationary time series, to be submitted in J. Math. Soc. Japan.
[13] Y. Okabe and T. Ootsuka, The theory of KM₂O-Langevin equations and its applications to data analysis (III): Prediction analysis, in preparation.
[14] J. von Neumann, Functional Operators, Vol. 2, Princeton, 1950.
[15] P. Whittle, On the fitting of multivariate autoregressions, and the approximate canonical factorization of a spectral density matrix, Biomerika, 50 (1963), 129-134.
[16] N. Wiener and P. Masani, The prediction theory of multivariate stochastic processes, I. The regularity condition, Acta Math., 98 (1957), 111-150.
[17] N. Wiener and P. Masani, The prediction theory of multivariate stochastic processes, II. The linear predictor, Acta Math., 99 (1958), 93-137.
[18] R. A. Wiggins and E. A. Robinson, Recursive solution to the multichannel fitting problem, J. Geophys. Res., 70 (1965), 1885-1891.

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