Journal of Statistical Physics

, Volume 24, Issue 2, pp 359–373 | Cite as

Linear systems and normality

  • Z. Kotulski
  • K. Sobczyk


This paper is concerned with responses of linear systems to non-Gaussian random excitation and with the measurement of the departure of the responses from Gaussian behavior. First, we show the classical Rosenblatt result and its nonapplicability to the most popular practical systems described by differential equations of first and second order. Then, using a simple measure of departure from normality (the asymmetry and excess coefficients) and performing numerical calculations, we give quantitative information about the effect of system parameters and the radius of correlation of the excitation process on the distance from normality.

Key words

Linear systems random excitation Gaussian processes limit theorems departure from normality 


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  1. 1.
    W. Hoeffing and H. Robbins,Duke Math. J. 15:773–780 (1948).Google Scholar
  2. 2.
    M. Rosenblatt,Proc. Nat. Acad. Sci. USA 42:43–47 (1956).Google Scholar
  3. 3.
    J. A. Rozanov,Probability Theory Appl. 5:243–246 (1960).Google Scholar
  4. 4.
    M. RosenblattQuart. Appl. Math. 18:387–393 (1961).Google Scholar
  5. 5.
    W. M. Wolkov,Electron. 15(12) (1970).Google Scholar
  6. 6.
    A. Papoulis,J. Appl. Probab. 8:118–127 (1971).Google Scholar
  7. 7.
    A. Papoulis,IEEE Trans. Inform. Theory IT-18(1):20–23 (1972).Google Scholar
  8. 8.
    C. L. Mallows,J. Appl Prob.,4:313–329 (1967).Google Scholar
  9. 9.
    K. Sobczyk, The Use of Gaussian Analysis in Non-Gaussian Problems, Department of Mechanical Engineering, Glasgow University (1976).Google Scholar
  10. 10.
    Z. Kotulski, Gaussianity in Stochastic Differential Equations (in Polish), M. Sci. thesis, Warsaw Technical University (1979).Google Scholar
  11. 11.
    J. L. Lumley, inStatistical Models and Turbulence, Lecture Notes in Physics (Springer, New York, 1972).Google Scholar
  12. 12.
    N. N. Kolmogorov and J. A. Rozanov,Probability Theory Appl. 5:222–227 (1960).Google Scholar
  13. 13.
    K. Sobczyk and D. B. Macvean inStochastic Problems in Dynamics (Pitman, New York, 1977).Google Scholar

Copyright information

© Plenum Publishing Corporation 1981

Authors and Affiliations

  • Z. Kotulski
    • 1
  • K. Sobczyk
    • 1
  1. 1.Institute of Fundamental Technological ResearchPolish Academy of SciencesWarsawPoland

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