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Stochastic Analysis and Related Topics pp 99–116Cite as

Birkhäuser

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

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

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References

  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)

    MathSciNet  MATH  Google Scholar 

  2. P. Billingsley, Convergence of Probability Measures, 2nd edn. (Wiley, New York, 1999)

    CrossRef  MATH  Google Scholar 

  3. P. Breuer, P. Major, Central limit theorems for nonlinear functionals of Gaussian fields. J. Multivar. Anal. 13(3), 425–441 (1983)

    CrossRef  MathSciNet  MATH  Google Scholar 

  4. C. Houdré, J. Villa, An example of infinite dimensional quasi-helix. Stoch. Models Contemp. Math. 366, 195–201 (2003)

    CrossRef  MathSciNet  MATH  Google Scholar 

  5. P. Lei, D. Nualart, A decomposition of the bifractional Brownian motion and some applications. Stat. Probab. Lett. 79, 619–624 (2009)

    CrossRef  MathSciNet  MATH  Google Scholar 

  6. I. Nourdin, G. Peccati, Normal Approximations with Malliavin Calculus: From Stein’s Method to Universality (Cambridge University Press, Cambridge, 2012)

    CrossRef  MATH  Google Scholar 

  7. D. Nualart, The Malliavin Calculus and Related Topics, 2nd edn. (Springer, Berlin, 2006)

    MATH  Google Scholar 

  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)

    CrossRef  MathSciNet  MATH  Google Scholar 

  9. G. Peccati, M.S. Taqqu, Wiener Chaos: Moments, Cumulants and Diagrams (Springer, Berlin, 2010)

    MATH  Google Scholar 

  10. V. Pipiras, M.S. Taqqu, Integration questions related to fractional Brownian motion. Probab. Theory Relat. Fields 118 251–291 (2000)

    CrossRef  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  13. C.A. Tudor, Analysis of Variations for Self-Similar Processes: A Stochastic Calculus Approach (Springer, Berlin, 2013)

    CrossRef  MATH  Google Scholar 

  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 

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Acknowledgements

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

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of Wisconsin Stevens Point, Stevens Point, WI, 54481, USA

    Daniel Harnett

  2. Department of Mathematics, University of Kansas, 405 Snow Hall, Lawrence, KS, 66045-2142, USA

    David Nualart

Authors
  1. Daniel Harnett
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  2. David Nualart
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Corresponding author

Correspondence to David Nualart .

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Connecticut, Storrs, Connecticut, USA

    Fabrice Baudoin

  2. Department of Mathematics, Purdue University, West Lafayette, Indiana, USA

    Jonathon Peterson

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Harnett, D., Nualart, D. (2017). Decomposition and Limit Theorems for a Class of Self-Similar Gaussian Processes. In: Baudoin, F., Peterson, J. (eds) Stochastic Analysis and Related Topics. Progress in Probability, vol 72. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-59671-6_5

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