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Decomposition and Limit Theorems for a Class of Self-Similar Gaussian Processes

  • Daniel Harnett
  • David Nualart
Conference paper
Part of the Progress in Probability book series (PRPR, volume 72)

Abstract

We introduce a new class of self-similar Gaussian stochastic processes, where the covariance is defined in terms of a fractional Brownian motion and another Gaussian process. A special case is the solution in time to the fractional-colored stochastic heat equation described in Tudor (Analysis of variations for self-similar processes: a stochastic calculus approach. Springer, Berlin, 2013). We prove that the process can be decomposed into a fractional Brownian motion (with a different parameter than the one that defines the covariance), and a Gaussian process first described in Lei and Nualart (Stat Probab Lett 79:619–624, 2009). The component processes can be expressed as stochastic integrals with respect to the Brownian sheet. We then prove a central limit theorem about the Hermite variations of the process.

Keywords

Fractional Brownian motion Hermite variations Self-similar processes Stochastic heat equation 

AMS 2010 Classification

60F05 60G18 60H07 

Notes

Acknowledgements

D. Nualart is supported by NSF grant DMS1512891 and the ARO grant FED0070445

References

  1. 1.
    R.M. Balan, C.A. Tudor, The stochastic heat equation with fractional-colored noise: existence of the solution. Latin Am. J. Probab. Math. Stat. 4, 57–87 (2008)MathSciNetMATHGoogle Scholar
  2. 2.
    P. Billingsley, Convergence of Probability Measures, 2nd edn. (Wiley, New York, 1999)CrossRefMATHGoogle Scholar
  3. 3.
    P. Breuer, P. Major, Central limit theorems for nonlinear functionals of Gaussian fields. J. Multivar. Anal. 13(3), 425–441 (1983)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    C. Houdré, J. Villa, An example of infinite dimensional quasi-helix. Stoch. Models Contemp. Math. 366, 195–201 (2003)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    P. Lei, D. Nualart, A decomposition of the bifractional Brownian motion and some applications. Stat. Probab. Lett. 79, 619–624 (2009)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    I. Nourdin, G. Peccati, Normal Approximations with Malliavin Calculus: From Stein’s Method to Universality (Cambridge University Press, Cambridge, 2012)CrossRefMATHGoogle Scholar
  7. 7.
    D. Nualart, The Malliavin Calculus and Related Topics, 2nd edn. (Springer, Berlin, 2006)MATHGoogle Scholar
  8. 8.
    H. Ouahhabi, C.A. Tudor, Additive functionals of the solution to the fractional stochastic heat equation. J. Fourier Anal. Appl. 19(4), 777–791 (2012)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    G. Peccati, M.S. Taqqu, Wiener Chaos: Moments, Cumulants and Diagrams (Springer, Berlin, 2010)MATHGoogle Scholar
  10. 10.
    V. Pipiras, M.S. Taqqu, Integration questions related to fractional Brownian motion. Probab. Theory Relat. Fields 118 251–291 (2000)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    G. Samorodnitsky, M.S. Taqqu, Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance (Chapman & Hall, New York, 1994)Google Scholar
  12. 12.
    S. Torres, C.A. Tudor, F. Viens, Quadratic variations for the fractional-colored stochastic heat equation. Electron. J. Probab. 19(76), 1–51 (2014)MathSciNetMATHGoogle Scholar
  13. 13.
    C.A. Tudor, Analysis of Variations for Self-Similar Processes: A Stochastic Calculus Approach (Springer, Berlin, 2013)CrossRefMATHGoogle Scholar
  14. 14.
    C.A. Tudor, Y. Xiao, Sample paths of the solution to the fractional-colored stochastic heat equation. Stochastics Dyn. 17(1) (2017)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of Wisconsin Stevens PointStevens PointUSA
  2. 2.Department of MathematicsUniversity of KansasLawrenceUSA

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