On Baxter Type Theorems for Generalized Random Gaussian Fields

  • Sergey KrasnitskiyEmail author
  • Oleksandr Kurchenko
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 271)


Some type of Baxter sums for generalized random Gaussian fields are introduced in this work. Sufficient conditions of such a sum convergence to a non-random constant are obtained. As the examples, the behavior of Baxter sums for a class of generalized fields with independent values and for a field of fractional Brownian motion is considered.


Levy–Baxter theorems Generalized random field Gaussian field 


  1. 1.
    Levy, P.: Le mouvement Brownian plan. Am. J. Math. 62, 487–550 (1940)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Baxter, G.: A strong limit theorem for Gaussian processes. Proc. Am. Math. Soc. 7, 522–527 (1956)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Gladyshev, E.G.: A new limit theorem for stochastic processes with Gaussian increments. Teor. Veroyatn. Primen. 6, 57–66 (1961)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Ryzhov, Yu.M.: One limit theorem for stationary Gaussian processes. Teor. Veroyatn. Mat. Stat. 1, 178–188 (1970)Google Scholar
  5. 5.
    Berman, S.M.: A version of the Levy-Baxter for the increments of Brownian motion of several parameters. Proc. Am. Math. Soc. 18, 1051–1055 (1967)Google Scholar
  6. 6.
    Krasnitskiy, S.M.: On some limit theorems for random fields with Gaussian m-order increments. Teor. Veroyatn. Mat. Stat. 5, 71–80 (1971)Google Scholar
  7. 7.
    Arak, T.V.: On Levy-Baxter type theorems for random fields. Teor. Veroyatn. Primen. 17, 153–160 (1972)Google Scholar
  8. 8.
    Kurchenko, O.O.: A strongly consistent estimate for the Hurst parameter of fractional Brownian motion. Teor. Imovir. Mat. Stat. 67, 45–54 (2002)Google Scholar
  9. 9.
    Breton, J.-C., Nourdin, I., Peccati, G.: Exact confidence intervals for the Hurst parameter of a fractional Brownian motion. Electron. J. Stat. 3, 416–425 (2009)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Kozachenko Yu.V., Kurchenko, O.O.: An estimate for the multiparameter FBM. Theory Stoch. Process 5(21), 113–119 (1999)Google Scholar
  11. 11.
    Prakasa Rao, B.L.S.: Statistical Inference for Fractional Diffusion Processes. Wiley, Chichester (2010)CrossRefGoogle Scholar
  12. 12.
    Gelfand, I.M., Vilenkin, N.Ya.: Applications of Harmonic Analysis. Equipped Hilbert Spaces. Fizmatgiz, Moscow (1961)Google Scholar
  13. 13.
    Rozanov, Yu.A.: Random Fields and Stochastic Partial Differential Equations. Nauka, Moscow (1995)Google Scholar
  14. 14.
    Goryainov, V.B.: On Levy-Baxter theorems for stochastic elliptic equations. Teor. Veroyatn. Primen. 33, 176–179 (1988)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Arato, N.M.: On a limit theorem for generalized Gaussian random fields corresponding to stochastic partial differential equations. Teor. Veroyatn. Primen. 34, 409–411 (1989)Google Scholar
  16. 16.
    Krasnitskiy, S.M., Kurchenko, O.O.: Baxter type theorems for generalized random Gaussian processes. Theory Stoch. Process. 21(37), 45–52 (2016)Google Scholar
  17. 17.
    Lamperti, J.: Stochastic Processes. A Survey of the Mathematical Theory. Vyscha shkola, Kyiv (1983)Google Scholar
  18. 18.
    Ibragimov, I.A., Rozanov, Y.A.: Gaussian Random Processes. Nauka, Moscow (1970)Google Scholar
  19. 19.
    Kamount, A.: On the fractional anisotropic random field. Probab. Math. Stat. 16, 85–98 (1996)Google Scholar

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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Kyiv National University of Technology and DesignKyivUkraine
  2. 2.Taras Shevchenko National University of KyivKyivUkraine

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