Skip to main content
SpringerLink
Log in

Limit theorem for the maximum of dependent Gaussian random elements in a Banach space

  • Brief Communications
  • Published:
Ukrainian Mathematical Journal Aims and scope

Abstract

The well-known Nisio result on the asymptotie equality for the maximum of real-valued Gaussian random variables is generalized to the case of Gaussian random variables taking values in a Banach space.

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

References

  1. M. Nisio, “On the extreme values of Gaussian processes,” Osaka J. Math. 4, 313–326 (1967).

    MATH  MathSciNet  Google Scholar 

  2. T. L. Lai, “Gaussian processes, moving averages and quick detection problems,” Ann. Probab., 1, No. 5, 825–837 (1973).

    Article  MATH  Google Scholar 

  3. I. K. Matsak, “Limit theorem for the maximum of Gaussian independent random variables in the space C,” Ukr. Mat. Zh., 47, No. 7, 1006–1008 (1995).

    Article  MATH  MathSciNet  Google Scholar 

  4. R. Carmona and N. Kono, “Convergence en loi et lois du logarithme itere pour les vecteurs gaussines,” Z. Wahrscheinlich. verw. Geb., 36, 241–267 (1976).

    Article  MATH  MathSciNet  Google Scholar 

  5. G.-C. Mangano, “On Strassen-type laws of the iterated logarithm for Gaussian elements in abstract spaces,” Z. Wahrscheinlich. verw. Geb., 36, 227–239 (1976).

    Article  MATH  MathSciNet  Google Scholar 

  6. M. Ledoux and M. Talagrand, Probability in Banach Spaces, Springer-Verlag, Berlin-Heidelberg (1991).

    MATH  Google Scholar 

  7. D. Bosq, “Properties asymptotiques des processus autogressifs banachiques. Applications,” C. R. Akad. Sci. Ser. l., 316, No. 6, 607–610 (1993).

    MATH  MathSciNet  Google Scholar 

  8. H. F. Chen and L. Guo, “Laws of iterated logarithm for state variables with applications,” Acta Math. Sin. New Ser., 4, No. 4, 343–355 (1988).

    Article  MATH  MathSciNet  Google Scholar 

  9. R. W. R. Darling, “Infinite-dimensional stochastic difference equations for particle systems and network flows,” J. Appl. Probab., 26, No. 2, 325–344 (1989).

    Article  MATH  MathSciNet  Google Scholar 

  10. V. A. Koval, “Laws of large numbers and repeated logarithm for solutions of stochastic difference equations in a Banach space,” in: Stochastic Equations and Limit Theorems [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1991), pp. 79–90.

    Google Scholar 

  11. V. A. Koval', “Loglog-principle of invariance for solutions of a stochastic difference equation in a Banach space,” in: Random Processes and Infinite-Dimensional Analysis [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1992), pp. 57–62.

    Google Scholar 

  12. A. Pechtl, “Arithmetic means and invariance principles in stochastic approximation,” J. Theor. Probab., 6, No. 1, 153–173 (1993).

    Article  MATH  MathSciNet  Google Scholar 

  13. V. A. Koval', “Weak invariance principle for solutions of a stochastic recurrence equation in a Banach space,” Ukr. Mat. Zh., 47, No. 1, 114–117 (1995).

    Article  MathSciNet  Google Scholar 

  14. V. Koval and R. Schwabe, Grenzwertsätze für Lösungen Stochastischer Differenzengleichungen in Banach-Räumen mit Anwendungen, Preprint No. A(32)95, Freie Univ. Berlin, Farchbereich Mathematik und Informatik, Berlin (1995).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Instrument-Constructing Institute, Zhitomir

    V. A. Koval’

  2. Free University, Berlin

    R. Schwabe

Authors
  1. V. A. Koval’
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. R. Schwabe
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 7, pp. 1005–1008, July, 1997.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Koval’, V.A., Schwabe, R. Limit theorem for the maximum of dependent Gaussian random elements in a Banach space. Ukr Math J 49, 1129–1133 (1997). https://doi.org/10.1007/BF02528760

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02528760

Keywords

Search

Navigation