Skip to main content
Log in

Supremum of the Euclidean Norms of the Multidimensional Wiener Process and Brownian Bridge: Sharp Asymptotics of Probabilities of Large Deviations

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


For T > 0, we prove theorems concerning sharp asymptotics of the probabilities

$$ \mathbf{P}\left\{\underset{t\in \left[0,T\right]}{\sup}\sum \limits_{j=1}^n{w}_j^2(t)>{u}^2\right\},\kern1em \mathbf{P}\left\{\underset{t\in \left[0,T\right]}{\sup}\sum \limits_{j=1}^n{w}_{j0,T}^2(t)>{u}^2\right\}, $$

as u → ∞, where wj(t), j = 1, . . . , n, are independent Wiener processes and wj0,T (t), j = 1, . . . , n, are independent Brownian bridges on the segment [0, T]. Our research method is the double sum method for the Gaussian processes and fields. We also give an application of the obtained results to the statistical tests for the homogeneity hypothesis of k one-dimensional samples.

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

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Similar content being viewed by others


  1. I. G. Abrahamson, “Exact Bahadur efficiencies for the Kolmogorov–Smirnov and Kuiper one- and two-sample statistics,” Ann. Math. Statist., 38, No. 5, 1475–1490 (1967).

    Article  MathSciNet  Google Scholar 

  2. Yu. K. Belyaev and V. I. Piterbarg, “The asymptotic behavior of the average number of the A-points of upcrossings of a Gaussian field beyond a high level,” in: Belyaev Yu. K., ed., Upcrossings of Random Fields [in Russian], Izd. Mosk. Univ., Moscow (1972), pp. 62–89.

    Google Scholar 

  3. A. N. Borodin and P. Salminen, Handbook of Brownian Motion [in Russian], Lan’, St.-Petersburg (2000).

  4. V. R. Fatalov, “Asymptotics of large deviations of Gaussian fields,” Izv. AN Armenii. Matem., 27, No. 6, 59–81. (1992).

    MathSciNet  Google Scholar 

  5. V. R. Fatalov, “Asymptotics of large deviations of Gaussian fields. Applications,” Izv. AN Armenii. Matem., 28, No. 5, 32–55. (1993).

    MathSciNet  Google Scholar 

  6. V. R. Fatalov, “Double sum method for the Gaussian fields with parameter set from lp space,” Fundam. Prikl. Matem., 2, No. 4, 1117–1141. (1996).

    MathSciNet  MATH  Google Scholar 

  7. V. R. Fatalov, “Large deviations for Gaussian processes in Hölder norm,” Izv. Math., 67, No. 5, 1061–1079 (2003).

    Article  MathSciNet  Google Scholar 

  8. V. R. Fatalov, “Occupation times and exact asymptotics of small deviations of Bessel processes for Lp-norms with p > 0,” Izv. Math., 71, No. 4, 721–752 (2007).

    Article  MathSciNet  Google Scholar 

  9. V. R. Fatalov, “Ergodic means for large values of T and exact asymptotics of small deviations for a multi-dimensional Wiener process,” Izv. Math., 77, No. 6, 1224–1259 (2013).

    Article  MathSciNet  Google Scholar 

  10. N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North Holland, Amsterdam (1981).

    MATH  Google Scholar 

  11. J. L. Kelly, General Topology, Springer, New York (1975).

    Google Scholar 

  12. J. Kiefer, “K-sample analogues of the Kolmogorov–Smirnov and Cramér–v. Mises tests,” Ann. Math. Statist., 30, No. 2, 420–447 (1959).

    Article  MathSciNet  Google Scholar 

  13. H. J. Landau and L. A. Shepp, “On the supremum of a Gaussian process,” Sankhyà, A32, No. 4, 369–378 (1971).

    MathSciNet  MATH  Google Scholar 

  14. M. Leadbetter, G. Lindgren, and H. Rootzen, Extremes and Related Properties of Random Sequences and Processes, Springer, New York (1983).

    Book  Google Scholar 

  15. M. A. Lifshits, Gaussian Random Functions, Kluwer, Dordrecht (1995).

    Book  Google Scholar 

  16. M. B. Marcus and L. A. Shepp, “Continuity of Gaussian processes,” Trans. Am. Math. Soc., 151, 377–392 (1970).

    Article  MathSciNet  Google Scholar 

  17. A. S. Mishchenko and A. T. Fomenko, A Course in Differential Geometry and Topology [in Russian], Izd. Mosk. Univ., Moscow (1980).

  18. Y. Nikitin, Asymptotic Efficiency of Nonparametric Tests, Cambridge Univ. Press, Cambridge (1995).

    Book  Google Scholar 

  19. A. A. Novikov, “Small deviations of Gaussian process,” Mat. Zametki, 29, 150–155 (1981).

    MATH  Google Scholar 

  20. V. Piterbarg, “High deviations for multidimensional stationary Gaussian process with independent coordinates,” in: Univ. of Lund and Lund Inst. of Technology, Dept. of Math. Statistics. Reports, Vol. 6, Lund (1991), pp. 1–34.

  21. V. I. Piterbarg, “High excursions for nonstationary generalized chi-square processes,” Stoch. Proc. Appl., 53, No. 2, 307–337 (1994).

    Article  MathSciNet  Google Scholar 

  22. V. I. Piterbarg, Asymptotic Methods in the Theory of Gaussian Processes and Fields, Amer. Math. Soc., Providence (2012).

    Book  Google Scholar 

  23. V. I. Piterbarg and V. R. Fatalov, “The Laplace method for probability measures in Banach spaces,” Russ. Math. Surv., 50, No. 6, 1151–1239 (1995).

    Article  MathSciNet  Google Scholar 

  24. D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, Springer, Berlin (1999).

    Book  Google Scholar 

  25. W. Rudin, Principles of Mathematical Analysis, McGraw-Hill, New York (1976).

    MATH  Google Scholar 

  26. S. M. Rytov, Introduction to Statistical Radiophysics [in Russian], Nauka, Moscow, (1966).

  27. B. Simon, Functional Integration and Quantum Physics, Academic Press, New York (1979).

    MATH  Google Scholar 

  28. A. D. Ventzel and M. I. Freidlin, Fluctuations in Dynamic Systems under the Action of Small Random Perturbations [in Russian], Nauka, Moscow (1979).

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to V. R. Fatalov.

Additional information

V. R. Fatalov is deceased.

Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 23, No. 1, pp. 219–257, 2020.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Fatalov, V.R. Supremum of the Euclidean Norms of the Multidimensional Wiener Process and Brownian Bridge: Sharp Asymptotics of Probabilities of Large Deviations. J Math Sci 262, 546–573 (2022).

Download citation

  • Published:

  • Issue Date:

  • DOI: