Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-26T14:39:42.348Z Has data issue: false hasContentIssue false
  • English
  • Français

Spectral characterization of the Wold–Zasuhin decomposition and prediction-error operator

Published online by Cambridge University Press:  24 October 2008

S. C. Power
S. C. Power
Affiliation:
Department of Mathematics, University of Lancaster, Lancaster LA 1 4YF
Get access Rights & Permissions [Opens in a new window]

Extract

Over thirty years ago Wiener and Masani pointed out in the introduction of their celebrated paper [31] that for a general multivariate stationary stochastic process no relation had been given for the prediction-error matrix in terms of the spectrum of the process. In particular it was unknown how to characterize the rank of the process in spectral terms (cf. Masani[12], p. 369, question 1). Despite explicit progress in this connection with certain regular processes, such as the series representations in [11, 19, 22, 32], or the iterative approach of [28, 29], and despite progress in the structure theory of degenerate processes ([8, 10, 14, 15, 26]), a general relation or series expression has remained elusive.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 110 , Issue 3 , November 1991 , pp. 559 - 567
Copyright
Copyright © Cambridge Philosophical Society 1991

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Arveson, W. B.. Interpolation problems in nest algebras. J. Funct. Anal. 20 (1975), 208–233.CrossRefGoogle Scholar
[2]Cuoi, M. D.. Some assorted inequalities for positive linear maps on C*-algebras. J. Operator Theory 4 (1980), 271–285.Google Scholar
[3]Devinatz, A.. The factorization of operator valued functions. Ann. of Math. 73 (1961), 458–459.CrossRefGoogle Scholar
[4]Doob, J. L.. Stochastic Processes (Wiley, 1953).Google Scholar
[5]Douglas, R. C.. On factoring positive operator functions. J. Math. Mech. 16 (1966), 119–126.Google Scholar
[6]Douglas, R. C.. Local Toeplitz operators. Proc. London Math. Soc. (3) 36 (1978), 243–472.CrossRefGoogle Scholar
[7]Helson, H.. Lectures on Invariant Subspaces (Academic Press, 1964).Google Scholar
[8]Helson, H. and Lowdernlager, D.. Prediction theory and Fourier series in several variables. Acta Math. 99 (1958), 165–201.CrossRefGoogle Scholar
[9]Lance, E. C.. Cohomology and perturbation of nest algebras. Proc. London Math. Soc. (3) 43 (1981), 334–356.Google Scholar
[10]Masani, P.. Cramer's theorem on monotone matrix-valued functions and the Wold decomposition. In Probability and Statistics (ed. Grenander, U.), pp. 175–189.Google Scholar
[11]Masani, P.. The prediction theory of multivariate stochastic processes III. Acta Math. 104 (1960), 141–162.CrossRefGoogle Scholar
[12]Masani, P.. Recent trends in multivariate prediction theory. In Multivariate Analysis (ed. Krishnaiah, P. R.) (Academic Press, 1966), pp. 351–382.Google Scholar
[13]Masani, P.. The place of multiplicative integration in modern analysis. Appendix II of Product Integration with Applications to Differential Equations, Encyclopedia of Mathematics and its Applications, vol. 10 (Addison Wesley, 1979).Google Scholar
[14]Matveev, R. F.. On multidimensional regular stationary processes. Theory Probab. Appl. 5 (1960), 33–39.Google Scholar
[15]Niemi, H.. Subordination, rank, and determinism of multivariate stationary sequences. J. Multivariate Anal.a 15 (1984), 99–123.CrossRefGoogle Scholar
[16]Page, L. B.. Bounded, compact and vectorial Hankel operators. Trans. Amer. Math. Soc. 150 (1970), 529–539.Google Scholar
[17]Peller, V. V.. Vectorial Hankel operators, commutators and related operators of the Schatten–von Neumann class Cp. J. Integral Equation. 5 (1982), 244–272.Google Scholar
[18]Peller, V. V. and Hruscev, S. V.. Hankel operators, best approximation and stationary Gaussian processes. Uspekhi Mat. Nauk. 37 (1982), 53–124.Google Scholar
[19]Poueahmadi, M.. A matricial extension of the Helson–Szego theorem and its application in multivariate prediction. J. Multivariate Anal. 16 (1965), 265–275.CrossRefGoogle Scholar
[20]Power, S. C.. Nuclear operators in nest algebras. J. Operator Theory 10 (1983), 337–352.Google Scholar
[21]Power, S. C.. Factorization in analytic operator algebras. J. Funct. Anal. 67 (1986), 413–432.CrossRefGoogle Scholar
[22]Rosanov, Y. A.. Stationary Random Processes (Holden Day, 1967).Google Scholar
[23]Rosenblum, M. and Rovnyak, J.. The factorization problem for non-negative operator-valued functions. Bull. Amer. Math. Soc. 77 (1971), 287–318.CrossRefGoogle Scholar
[24]Sarason, D.. Functions of vanishing mean oscillations. Trans. Amer. Math. Soc. 207 (1973), 286–299.Google Scholar
[25]Smul'jan, Ju. L.. An operator Hellinger integral. Mat. Sb. 91 (1959), 381–430.Google Scholar
[26]Suciu, I. and Valusescu, I.. Factorization theorems and prediction theory. Rev. Roumaine Math. Pures Appl. 23 (1978), 1393–1423.Google Scholar
[27]Sz-Nagy, B. and Fotaş, C.. Harmonic Analysis of Hilbert Space Operators (North Holland, 1970).Google Scholar
[28]Wilson, G. Tunnicliffe. The factorization of matricial spectral densities. SIAM. J. Appl. Math. 23 (1972), 420–426.CrossRefGoogle Scholar
[29]Wilson, G. Tunnicliffe. A convergence theorem for spectral factorization. J. Multivariate Anal. 8 (1978), 222–232.CrossRefGoogle Scholar
[30]Wiener, N. and Akutowicz, E. J.. A factorization of positive hermitian matrices. J Math. Mech. 8 (1959), 111–120.Google Scholar
[31]Wiener, N. and Masani, P.. The prediction theory of multivariate stochastic processes. Part I, Acta Math. 98 (1957), 111–150.CrossRefGoogle Scholar
[32]Wiener, N. and Masani, P.. The prediction theory of multivariate stochastic processes. Part II, Acta Math. 99 (1958), 93–137.CrossRefGoogle Scholar