Skip to main content
SpringerLink
Log in

Multiparameter Gaussian Processes

  • Chapter
Analysis of Variations for Self-similar Processes

Part of the book series: Probability and Its Applications ((PIA))

  • 1462 Accesses

Abstract

Chapter 4 treats multiparameter self-similar processes. The main focus is on the fractional Brownian sheet and on the multiparameter Hermite processes. We describe various properties of these processes such as the self-similarity, the stationarity of the increments, continuity or their stochastic integral representation.

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

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 119.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 119.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. R.J. Adler, The Geometry of Random Fields (Wiley, New York, 1981)

    MATH  Google Scholar 

  2. A. Ayache, S. Leger, M. Pontier, Drap Brownien fractionnaire. Potential Anal. 17(1), 31–43 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  3. X. Bardina, M. Jolis, C. Rovira, Weak approximation of a Wiener process from a Poisson process: the multidimensional parameter set case. Stat. Probab. Lett. 50(3), 245–255 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  4. X. Bardina, M. Jolis, C.A. Tudor, Weak convergence to the fractional Brownian sheet and other two-parameter Gaussian processes. Stat. Probab. Lett. 65(4), 317–329 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  5. A. Bonami, A. Estrade, Anisotropic analysis of some Gaussian models. J. Fourier Anal. Appl. 9(3), 215–236 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  6. J.-C. Breton, On the rate of convergence in non-central asymptotics of the Hermite variations of fractional Brownian sheet. Probab. Math. Stat. 31(2), 301–311 (2011)

    MathSciNet  MATH  Google Scholar 

  7. N.N. Cenkov, Wiener random fields depending on several parameters. Dokl. Akad. Nauk SSSR 106, 607–609 (1956)

    MathSciNet  Google Scholar 

  8. Z. Ciesielski, A. Kamont, Lévy’s fractional Brownian random field and function spaces. Acta Sci. Math. 60, 99–118 (1995)

    MathSciNet  MATH  Google Scholar 

  9. J. Clarke De la Cerda, C.A. Tudor, Least squares estimator for the parameter of the fractional Ornstein-Uhlenbeck sheet. J. Korean Stat. Soc. 41(3), 341–350 (2012)

    Article  Google Scholar 

  10. J. Clarke De la Cerda, C.A. Tudor, Wiener integrals with respect to the Hermite random field and applications to the wave equation. Preprint (2013)

    Google Scholar 

  11. M. Dozzi, Stochastic Processes with a Multidimensional Parameter. Pitman Research Notes in Mathematics Series, vol. 194 (Longman Scientific and Technical, Harlow, 1989)

    MATH  Google Scholar 

  12. M. Eddahbi, R. Lacayo, J.L. Sole, C.A. Tudor, J. Vives, Regularity of the local time for the d-dimensional fractional Brownian motion with N-parameters. Stoch. Anal. Appl. 23(2), 383–400 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  13. M. Erraoui, D. Nualart, Y. Ouknine, Hyperbolic stochastic partial differential equations with additive fractional Brownian sheet. Stoch. Dyn. 3(2), 121–139 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  14. A. Kamont, On the fractional anisotropic Wiener field. Probab. Math. Stat. 16(1), 85–98 (1996)

    MathSciNet  MATH  Google Scholar 

  15. D. Khosnevisan, Multiparameter Stochastic Processes (Springer, New York, 2002)

    Google Scholar 

  16. T. Lindstrom, Fractional Brownian fields as integrals of white noise. Bull. Lond. Math. Soc. 25(1), 83–88 (1993)

    Article  MathSciNet  Google Scholar 

  17. D. Nualart, Malliavin Calculus and Related Topics, 2nd edn. (Springer, New York, 2006)

    MATH  Google Scholar 

  18. A. Réveillac, M. Stauch, C.A. Tudor, Hermite variations of the fractional Brownian sheet. Stoch. Dyn. 12(3), 1150021 (2012), 21 pp.

    Article  MathSciNet  Google Scholar 

  19. A. Réveillac, Estimation of quadratic variation for two-parameter diffusions. Stoch. Process. Appl. 119(5), 1652–1672 (2009)

    Article  MATH  Google Scholar 

  20. A. Réveillac, Convergence of finite-dimensional laws of the weighted quadratic variations process for some fractional Brownian sheets. Stoch. Anal. Appl. 27(1), 51–73 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  21. C.A. Tudor, F. Viens, Itô formula and local time for the fractional Brownian sheet. Electron. J. Probab. 8(14), 31 (2003)

    MathSciNet  Google Scholar 

  22. C.A. Tudor, F. Viens, Itô formula for the two-parameter fractional Brownian motion using the extended divergence operator. Stochastics 78(6), 443–462 (2006)

    MathSciNet  MATH  Google Scholar 

  23. C. Tudor, M. Tudor, On the multiparameter fractional Brownian motion, in Advances in Mathematics Research. Adv. Math. Res., vol. 6 (Nova Sci. Publ., New York, 2005), pp. 121–140

    Google Scholar 

  24. Y. Xiao, Sample path properties of anisotropic Gaussian random fields, in A Minicourse on Stochastic Partial Differential Equations. Lecture Notes in Math., vol. 1962 (Springer, Berlin, 2009), pp. 145–212

    Chapter  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. UFR Mathématiques, Université de Lille 1, Villeneuve d’Ascq, France

    Ciprian A. Tudor

Authors
  1. Ciprian A. Tudor
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Tudor, C.A. (2013). Multiparameter Gaussian Processes. In: Analysis of Variations for Self-similar Processes. Probability and Its Applications. Springer, Cham. https://doi.org/10.1007/978-3-319-00936-0_4

Download citation

Publish with us

Policies and ethics

Search

Navigation