Skip to main content
SpringerLink
Log in

On the properties of an empirical correlogram of a Gaussian process with square integrable spectral density

  • Published:
Ukrainian Mathematical Journal Aims and scope

Abstract

We study properties of an empirical correlogram of a centered stationary Gaussian process. We prove that if the spectral density of the process is square integrable, then there is a normalization effect for the correlogram and integral functionals of it.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. I. A. Ibragimov, “On an estimate of the spectral function of a stationary Gaussian process,”Teor. Ver. Primen.,8, No. 4, 391–430 (1963).

    Google Scholar 

  2. R. Bentkus, “Asymptotic normality of an estimate of a spectral function.”Lit. Mat. Sb.,12, No. 3, 3–17 (1972).

    Google Scholar 

  3. A. V. Ivanov, “A limit theorem for the estimation of a correlation function,”Teor. Ver. Mat. Stat., Issue 19, 76–81 (1978).

    Google Scholar 

  4. A. V. Ivanov and N. N. Leonenko,Statistical Analysis of Random Fields [in Russian], Vyshcha Shkola, Kiev (1983),English translation: Kluwer, Dordrecht-Boston-London (1989).

    Google Scholar 

  5. A. A. Dykhovichnyi, “On an estimate of the correlation function of a homogeneous isotropic Gaussian field,”Teor. Ver. Mat. Stat., Issue 29, 37–40(1983).

    Google Scholar 

  6. V. V. Buldygin and V. V. Zayats, “Asymptotic normality of an estimate of the correlation in different functional spaces,” in:Probability Theory and Mathematical Statistics, World Scientific, Singapore (1992), pp. 19–31.

    Google Scholar 

  7. Yu. V. Kozachenko and A. I. Stadnik, “Pre-Gaussian processes and the rate of convergence of estimates of covariance functions inC(T),”Teor. Ver. Mat Stat., Issue 45, 54–62 (1991).

  8. V. V. Buldygin and O. O. Dem'yanenko, “Asymptotic normality of an estimate of a joint correlation function in functional spaces,”Dokl. Akad. Nauk Ukr., Ser. A, No. 3, 32–36 (1993).

    Google Scholar 

  9. N. N. Leonenko and A. Yu. Portnova, “Convergence of the correlogram of a Gaussian field to a non-Gaussian field,”Teor. Ver. Mat. Stat., Issue 49, 137–144 (1993).

    Google Scholar 

  10. V. P. Leonov and A. N. Shiryaev, “Method for calculation of semiinvariants,”Teor. Ver. Primen.,4, No. 3, 342–355 (1959).

    Google Scholar 

  11. R. E. Edwards,Fourier Series. A Modern Introduction, Vol. 1, Springer, New York-Heidelberg-Berlin (1982).

    Google Scholar 

  12. P. Billingsley,Convergence of Probability Measures, Wiley, New York (1968).

    Google Scholar 

  13. L. Ŝ. Grinblat, “A limit theorem for measurable random processes and its applications,”Proc. Am. Math. Soc.,61, No. 2., 371–376 (1976).

    Google Scholar 

  14. A. A. Borovkov and E. A. Pecherskii, “Convergence of distributions of integral functionals,”Sib. Mat. Zh.,16, No. 5, 899–915 (1975).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Kiev Polytechnic Institute, Kiev

    V. V. Buldygin

Authors
  1. V. V. Buldygin
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 7, pp. 876–889, July, 1995.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Buldygin, V.V. On the properties of an empirical correlogram of a Gaussian process with square integrable spectral density. Ukr Math J 47, 1006–1021 (1995). https://doi.org/10.1007/BF01084897

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01084897

Keywords

Search

Navigation