Skip to main content
SpringerLink
Log in

On Global and Local Properties of the Trajectories of Gaussian Random Fields—A Look Through the Set of Limit Points

  • Published:
Acta Mathematica Sinica, English Series Aims and scope Submit manuscript

Abstract

This paper studies the global and local properties of the trajectories of Gaussian random fields with stationary increments and proves sufficient conditions for Strassen’s functional laws of the iterated logarithm at zero and infinity respectively. The sets of limit points of those Gaussian random fields are obtained. The main results are applied to fractional Riesz–Bessel processes and the sets of limit points of this field are obtained.

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. Addie, R., Mannersalo, P., Norros, I.: Most probable paths and performance formulae for buffers with Gaussian input traffic. European Trans. Telecommunications, 13, 183–196 (2002)

    Article  Google Scholar 

  2. Anh, V. V., Angulo, J. M., Ruiz-Medina, M. D.: Possible long-range dependence in fractional random fields. J. Statist. Plann. Inference, 80, 95–110 (1999)

    Article  MathSciNet  Google Scholar 

  3. Arcones, M. A.: The large deviation principle for stochastic processes, Part II. Theoret. Probab. Appl., 48, 19–44 (2003)

    Article  Google Scholar 

  4. Aronszajn, N.: Theory of reproducing kernels. Trans. Amer. Math. Soc., 68, 337–404 (1950)

    Article  MathSciNet  Google Scholar 

  5. Benson, D. A., Meerschaert, M. M., Baeumer, B.: Aquifer operator-scaling and the effect on solute mixing and dispersion. Water Resour. Res., 42, W01415 (2006)

    Article  Google Scholar 

  6. Berg, C., Forst, G.: Potential Theory on Locally Compact Abelian Groups, Springer-Verlag, New York-Heidelberg, 1975

    Book  Google Scholar 

  7. Biermé, H., Meerschaert, M. M., Scheffler, H. P.: Operator scaling stable random fields. Stoch. Process. Appl., 117, 312–332 (2007)

    Article  MathSciNet  Google Scholar 

  8. Bonami, A., Estrade, A.: Anisotropic analysis of some Gaussian models. J. Fourier Anal. Appl., 9, 215–236 (2003)

    Article  MathSciNet  Google Scholar 

  9. Cheridito, P.: Gaussian moving averages, semimartingales and option pricing. Stoch. Process. Appl., 109, 47–68 (2004)

    Article  MathSciNet  Google Scholar 

  10. Cressie, N. A. C.: Statistics for Spatial Data, Revised edition, John Wiley & Sons, Inc., New York, 2015

    MATH  Google Scholar 

  11. Dudley, R. M.: Sample functions of the Gaussian processes. Ann. Probab., 3, 66–103 (1973)

    Article  Google Scholar 

  12. Gneiting, T., Schlather, M.: Stochastic models that separate fractal dimension and the Hurst effect. SIAM Review, 46, 269–282 (2004)

    Article  MathSciNet  Google Scholar 

  13. Khoshnevisan, D.: Multiparameter Processes: An Introduction to Random Fields, Springer-Verlag, New York, 2002

    Book  Google Scholar 

  14. Mannersalo, P., Norros, I.: A most probable path approach to queueing systems with general Gaussian input. Comp. Networks, 40, 399–412 (2002)

    Article  Google Scholar 

  15. Marcus, M. B., Rosen, J.: Markov Processes, Gaussian Processes, and Local Times, Cambridge University Press, Cambridge, 2006

    Book  Google Scholar 

  16. Meerschaert, M. M., Wang, W., Xiao, Y.: Fernique type inequality and moduli of continuity for anisotropic Gaussian random fields. Trans. Amer. Math. Soc., 365, 1081–1107 (2013)

    Article  MathSciNet  Google Scholar 

  17. Mueller, C., Tribe, R.: Hitting probabilities of a random string. Electronic J. Probab., 7, 1–29 (2002)

    Article  MathSciNet  Google Scholar 

  18. Samorodnitsky, G., Taqqu, M. S.: Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance, Chapman & Hall, New York, 1994

    MATH  Google Scholar 

  19. Talagrand, M.: Hausdorff measure of trajectories of multiparameter fractional Brownian motion. Ann. Probab., 23, 767–775 (1995)

    Article  MathSciNet  Google Scholar 

  20. Wang, W., Su, Z., Xiao, Y.: The Csörgő-Révész moduli of non-differentiability for Gaussian random fields. Preprint, 2018

    Chapter  Google Scholar 

  21. Xiao, Y.: Strong local nondeterminism and sample path properties of Gaussian random fields, In: Asympotic Theory in Probability and Statistics with Applications, T. L. Lai, Q. M. Shao and L. Qian, eds., International Press, Boston, 2006

    Google Scholar 

  22. Yaglom, A. M.: Some classes of random fields in n-dimensional space, related to stationary random processes. Theory Probab. Appl., 2, 273–320 (1957)

    Article  Google Scholar 

Download references

Acknowledgements

The authors wish to express their deep gratitude to the referees for their valuable comments on an earlier version which improve the quality of this paper.

Author information

Authors and Affiliations

  1. School of Economics, Hangzhou Dianzi University, Hangzhou, 310018, P. R. China

    Wen Sheng Wang

  2. School of Mathematical Sciences, Zhejiang University, Hangzhou, 310027, P. R. China

    Zhong Gen Su

  3. Department of Statistics and Probability, Michigan State University, East Lansing, MI, 48824, USA

    Yi Min Xiao

Authors
  1. Wen Sheng Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhong Gen Su
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yi Min Xiao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Wen Sheng Wang, Zhong Gen Su or Yi Min Xiao.

Additional information

Supported by NSFC (Grants Nos. 11671115, 11731012 and 11871425) and NSF (Grant No. DMS-1855185)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, W.S., Su, Z.G. & Xiao, Y.M. On Global and Local Properties of the Trajectories of Gaussian Random Fields—A Look Through the Set of Limit Points. Acta. Math. Sin.-English Ser. 36, 137–152 (2020). https://doi.org/10.1007/s10114-020-9083-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10114-020-9083-0

Keywords

MR(2010) Subject Classification

Search

Navigation