Skip to main content
SpringerLink
Log in

On the Sharp Constant in the Small Ball Asymptotics of Some Gaussian Processes under L 2-Norm

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

We find the sharp constant in the small L 2-deviation asymptotics for a wide class of Gaussian processes including the m-times integrated Wiener process and the m-times integrated Ornstein–Uhlenbeck process. Extremal properties of usual and Euler integration are proved. Bibliography: 19 titles.

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. M. A. Lifshits, “Asymptotic behavior of small ball probabilities,” in: Prob. Theory Math. Stat. (1999), pp. 453–468

  2. W. V. Li and Q. M. Shao, “Gaussian processes: Inequalities, Small Ball Probabilities and Applications,” in:Stochastic Processes: Theory and Methods, Handbook of Statistics, Vol. 19(2001)

  3. W. V. Li, “Comparison results for the lower tail of Gaussian seminorms,” J. Theor. Probab., 5, No. 1, 1–31(1992)

    Google Scholar 

  4. T. Dunker, M. A. Lifshits, and W. Linde, “Small deviations of sums of independent variables,” in: Proc. Conf. High Dimensional Probability. Ser. Progress in Probability, 43, 59–74(1998)

    Google Scholar 

  5. A. I. Nazarov and Ya. Yu. Nikitin, “Exact small ball behavior of integrated Gaussian processes under L2-norm and spectral asymptotics of boundary value problems,” Stud. Statist., No. 70, (2003)

  6. F. Gao, J. Hannig, and F. Torcaso, “Integrated Brownian motions and exact L2-small balls,” Colorado State University. Preprint (2001)

  7. I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Sums, Series and Products [in Russian],Nauka, Moscow (1971)

    Google Scholar 

  8. M. A. Naimark, Linear Differential Operators [in Russian], Nauka, Moscow (1969)

    Google Scholar 

  9. E. C. Titchmarsh, The Theory of Functions, Oxford University Press (1975)

  10. A. Lachal, “Study of some new integrated statistics: computation of Bahadur efficiency, relation with non-standard boundary value problems,” Math. Methods Stat., 10, No. 1, 73–104(2001)

    Google Scholar 

  11. A. Lachal, “Bridges of certain Wiener integrals. Prediction properties, relation with polynomial interpolation and differential equations. Application to goodness-of-fit testing,” in: Bolyai Math. Stud. X. Budapest (2002), pp. 1–51

  12. A.N. Borodin and P. Salminen, Handbook of Brownian Motion:Facts and Formulae, Birkh¨ auser, Basel (1996)

    Google Scholar 

  13. Ya. Yu. Nikitin, “Limit distributions and comparative asymptotic efficiency of the Kolmogorov–Smirnov statistics with a random index,” Zap. Nauchn. Semin. LOMI [in Russian], 55, 185–194(1976)

    Google Scholar 

  14. Ya. Yu. Nikitin and E. Orsingher, “The intermediate arc-sine law,” Stat. Probab. Letters, 49, 119–125(2000)

    Google Scholar 

  15. T. W. Anderson and D. A. Darling, “Asymptotic theory of certain “goodness-of-fit” criteria based on stochastic processes,” Ann. Math. Stat. 23, No. 1, 193–212(1952)

    Google Scholar 

  16. I. Norros, E. Valkeila, and J. Virtamo, “An elementary approach to a Girsanov formula and other analytical results on fractional Brownian motions,” Bernoulli, 5(4), 571–587(1999)

    Google Scholar 

  17. Functional Analysis [in Russian], Nauka, Moscow (1972)

  18. F. Gao, J. Hannig, T.-Y. Lee, and F. Torcaso, “Exact L 2 small balls of Gaussian processes,” J. Theor. Probab. [to appear]

  19. F. Gao, J. Hannig, T.-Y. Lee, and F. Torcaso, “Laplace Transforms via Hadamard Factorization with applica-tions to Small Ball probabilities,” [to appear]

Download references

Authors
  1. A.I. Nazarov
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Nazarov, A. On the Sharp Constant in the Small Ball Asymptotics of Some Gaussian Processes under L 2-Norm. Journal of Mathematical Sciences 117, 4185–4210 (2003). https://doi.org/10.1023/A:1024868604219

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1024868604219

Keywords

Search

Navigation