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On the sufficient equivalence conditions of gaussian measures corresponding to homogeneous fields whose spectral densities have real zeros

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Abstract

The sufficient equivalence criteria of probability measures corresponding to Gaussian homogeneous random fields are given in terms of the spectral densities of the fields considered. These criteria are represented in a form taking into account the possibility of vanishing of the mentioned spectral densities.

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References

  1. I.I. Gikhman and A. V. Skorokhod, Theory of Random Processes [in Russian], Vol. 1, Nauka, Moscow (1975).

    MATH  Google Scholar 

  2. M. I. Yadrenko, Spectral Theory of Random Fields [in Russian], Vyshcha Shkola, Kiev (1980).

    Google Scholar 

  3. I. M. Gel’fand and S. Ya. Vilenkin, Some Applications of Fourier Analysis: Framed Hubert Spaces [in Russian], Fizmatgiz, Moscow (1961).

    Google Scholar 

  4. A. M. Yaglom, “Some classes of random fields in an /i-dimensional space that are related to stationary random processes,” Teor. Veroyatn. Primen.,11, No. 3, 292–338 (1957).

    MathSciNet  Google Scholar 

  5. Yu. A. Rosanov, Random Fields and Stochastic Equations with Partial Derivatives [in Russian], Nauka, Moscow (1995).

    Google Scholar 

  6. V. S. Vladimirov, Generalized Functions in Mathematical Physics [in Russian], Nauka, Moscow (1976).

    Google Scholar 

  7. L. Hoermander, The Analysis of Linear Partial Differential Operators: Distribution Theory and Fourier Analysis [Russian translation], Vol. 1, Mir, Moscow (1986).

    MATH  Google Scholar 

  8. D. S. Apokorin, “Gaussian measures corresponding to generalized stationary processes,” Teor. Veroyatn. Primen., No. 4, 698–707 (1967).

  9. Yu. A. Rosanov, “Gaussian infinite-dimensional distributions,” Trans, of the V. A. Steklov Mat. Inst., Nauka, Moscow (1968).

    Google Scholar 

  10. A. V. Skorokhod and M. I. Yadrenko, “Absolute continuity of measures corresponding to homogeneous random fields,” Teor. Veroyatn. Primen., 18, No. 2, 30–43 (1973).

  11. S. M. Krasnitskii, “Equivalence conditions of distributions of Gaussian homogeneous fields,” in: Proc. of the 4th Intern. Conf. on the Theory of Probability and Mathematical Statistics, Vol. 2, Inst. Mat. Kibern. AN LitSSR, Vilnius (1985), pp. 72–74.

    Google Scholar 

  12. Z. S. Zerakidze, “On equivalence of distributions of Gaussian homogeneous fields,” in: Tr. Inst. Prikladn. Mat. Tbil. Univ., 2, Tbilisi (1969), pp. 215–220.

    Google Scholar 

  13. S. M. Krasnitskii, “A class of spectral equivalence conditions of Gaussian measures corresponding to homogeneous random fields,” Dop. NAN Ukr., No. 3, 37–39 (1993).

  14. J. Feldman, “Some classes of equivalent Gaussian processes on an interval,” Pacif. J. Math., 10, No. 4, 1211–1220 (1960).

  15. S. M. Krasnitskii, “A spectral equivalence condition of Gaussian measures,” in: Proc. of the Intern. Conf. Dedicated to the Memory of Hans Gan, Ruta, Chernovtsy (1994), p. 75.

    Google Scholar 

  16. S. M. Krasnitskii, “Equivalence conditions of distributions of Gaussian homogeneous fields,” Teor. Veroyatn. Primen., 34, No. 4, 786–791 (1989).

  17. L. Hoermander, Analysis of Linear Partial Differential Operators: Differential Operators with Constant Coefficients, [Russian translation], Vol. 2, Mir, Moscow (1986).

    MATH  Google Scholar 

  18. I. M. Gel’fand and G. E. Shilov, Spaces of Basic and Generalized Functions [in Russian], Fizmatgiz, Moscow (1958).

    Google Scholar 

  19. O. V. Besov, V. P. II’ in, and S. M. Nikorskii, Integral Representations of a Function and Embedding Theorems [in Russian], Nauka, Moscow (1975).

    Google Scholar 

  20. G. E. Shilov, Mathematical Analysis: A Second Special Course [in Russian], Nauka. Moscow (1965).

    Google Scholar 

  21. S. M. Krasnitskii, “On the necessary equivalence conditions of Gaussian measures corresponding to homogeneous random fields,” Ukr. Mat. Zh., 46, No. 6, 708–719 (1994).

    Google Scholar 

  22. S. M. Krasnitskii, “On the necessary and sufficient equivalence conditions of the probabilistic measures corresponding to homogeneous random fields having spectral densities,” Kibern. Sist. Anal., No. 5, 37–44 (1997).

    Google Scholar 

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Authors and Affiliations

  1. State Academy of Light Industry, Kiev, Ukraine

    S. M. Krasnitskii

  2. Cybernetics Institute, National Academy of Sciences of Ukraine, Kiev, Ukraine

    S. L. Kryvyi

Authors
  1. S. M. Krasnitskii
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  2. S. L. Kryvyi
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Translated from Kibernetika i Sistemnyi Analiz, No. 6, pp. 37–48, November–December, 1999.

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Krasnitskii, S.M., Kryvyi, S.L. On the sufficient equivalence conditions of gaussian measures corresponding to homogeneous fields whose spectral densities have real zeros. Cybern Syst Anal 35, 875–883 (1999). https://doi.org/10.1007/BF02742278

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  • DOI: https://doi.org/10.1007/BF02742278

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