Stationary Gaussian Processes Satisfying the Strong Mixing Condition and Best Predictable Functionals

  • A. M. Yaglom

Abstract

The well-known problem of least squares prediction of the stationary stochastic process x (t) is the problem of finding the functional \(\tilde{x}\text{ }\left( \tau \right)\) of the values x(t), t≤0, which is the least squares approximation of the “future” value of the process x(τ), τ>0. In addition to the functional \(\tilde{x}\text{ }\left( \tau \right)\) it is important to know the mean square error of the prediction
$${{\sigma }^{2}}\left( \tau \right)\text{ = E}{{\left| x\left( \tau \right)-\tilde{x}\left( \tau \right) \right|}^{2}},$$
(1)
or, what is equivalent, the correlation coefficient
$$\ell \left( \tau \right)=\frac{Ex\left( \tau \right)\tilde{x}\left( \tau \right)}{{{\left\{ E{{[x\left( \tau \right)]}^{2}}E{{[\tilde{x}\left( \tau \right)]}^{2}} \right\}}^{\frac{1}{2}}}}=\sqrt{1-\frac{{{\sigma }^{2}}\left( \tau \right)}{E{{[x\left( \tau \right)]}^{2}}}}$$
(2)
between x (τ) and \(\tilde{x}\left( \tau \right)\) [here and later we can consider without loss of generality only the processes x (t) with Ex (t) = 0]. If the process x (t) is Gaussian, the least squares approximation \(\tilde{x}\left( \tau \right)\) is linear; therefore, we can say that the problem of linear least squares prediction of the stationary process x (t) is the wide sense version of the general problem of least squares prediction (see DooB [1],Chapter II, Section 3).

Keywords

Manifold Covariance Dition 

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References

  1. [1]
    Doos, J. L.: Stochastic Processes. New York: Wiley 1953.Google Scholar
  2. [2]
    Kolmogorov, A. N.: Interpolation und Extrapolation von stationären zufälligen Folgen. Izv. Akad. Nauk SSSR. Ser. Mat. 5, 3 (1941).Google Scholar
  3. [3]
    Krein, M. G.: On a problem of extrapolation of A. N. Kolmogoroff. Dokl. A.ad. Nauk SSSR 46, 306 (1945).Google Scholar
  4. [4]
    Wiener, N.: Extrapolation, Interpolation, and Smoothing of Stationary Time Series. New York: Wiley 1949.Google Scholar
  5. [5]
    Yaglom, A. M.: An Introduction to the Theory of Stationary Random Functions. New York: Prentice-Hall 1962.Google Scholar
  6. [6]
    Rosenblatt, M.: A central limit theorem and a strong mixing condition. Proc. Nat. Acad. Sci. USA. 42, 43 (1956).CrossRefGoogle Scholar
  7. [7]
    Kolmogorov, A. N., and Yu. A. Rozanov: On a strong mixing condition for a stationary random Gaussian process. Teor. Veroyatnost. i Primenen. 5, 222 (1960).Google Scholar
  8. [8]
    Ibragimov, I. A.: Spectral functions of certain classes of Gaussian stationary processes. Dokl. Akad. Nauk SSSR. 137, 1046 (1961).Google Scholar
  9. [9]
    Ibragimov, I. A.: Stationary Gaussian sequences that satisfy the strong mixing condition. Dokl. Akad. Nauk SSSR. 147, 1282 (1962).Google Scholar
  10. [10]
    Rosenblatt, M.: Independence and dependence. Proc. Fourth Berkeley Symp. on Math. Stat. and Prob. II, 431–443. Berkeley and Los Angeles: University of California Press 1961.Google Scholar
  11. [11]
    Helson, H., and G. Szegö: A problem in prediction theory. Ann. Matem. Pura ed Appl. 51, 107 (1960).CrossRefGoogle Scholar
  12. [12]
    Hotelling, H.: Relation between two sets of variates. Biometrica. 28, 321 (1936).Google Scholar
  13. [13]
    Oboukhov, A. M.: Normal correlation of vectors. Izv. Akad. Nauk SSSR. Ser. Mat. 3, 339 (1938).Google Scholar
  14. [14]
    Oboukhov, A. M.: Theory of correlation of vectors. Uchen. Zap. Mosk. Gosud. Univ., Matem. 45, 73 (1940).Google Scholar
  15. [15]
    Anderson, T. W.: An Introduction to Multivariate Statistical Analysis. New York: Wiley 1959.Google Scholar
  16. [16]
    Gelfand, I. M., and A. M. Yaglom: Calculation of the amount of information about a random function contained in another such function. Uspehi Mat. Nauk. 12, No. 1 (73), 3 (1957).Google Scholar
  17. [17]
    Hannan, E. J.: The general theory of canonical correlation and its relation to functional analysis. J. Austral. Math. Soc. 2, 229 (1961).CrossRefGoogle Scholar
  18. [18]
    Shirokov, P. A.: Tensor Calculus. Moskva-Leningrad 1934.Google Scholar
  19. [19]
    Doobs, J. L.: The elementary Gaussian processes. Ann. Math. Statist. 15, 229 (1944).CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin · Heidelberg 1965

Authors and Affiliations

  • A. M. Yaglom
    • 1
  1. 1.Academy of Sciences of the USSRRussia

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