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Central Limit Theorems for Multiple Skorokhod Integrals

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Abstract

In this paper, we prove a central limit theorem for a sequence of multiple Skorokhod integrals using the techniques of Malliavin calculus. The convergence is stable, and the limit is a conditionally Gaussian random variable. Some applications to sequences of multiple stochastic integrals, and renormalized weighted Hermite variations of the fractional Brownian motion are discussed.

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References

  1. Baldi, P., Kerkyacharian, G., Marinucci, D., Picard, D.: Subsampling needlet coefficients on the sphere. Bernoulli 15(2), 438–463 (2009)

    Article  MathSciNet  Google Scholar 

  2. Barndorff-Nielsen, O., Corcuera, J., Podolskij, M., Woerner, J.: Bipower variations for Gaussian processes with stationary increments. J. Appl. Probab. 46(1), 132–150 (2009)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  4. Corcuera, J.M., Nualart, D., Woerner, J.H.C.: Power variation of some integral fractional processes. Bernoulli 12, 713–735 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  5. Decreusefond, L., Üstünel, A.S.: Stochastic analysis of the fractional Brownian motion. Potential Anal. 10, 177–214 (1998)

    Article  Google Scholar 

  6. Deheuvels, P., Peccati, G., Yor, M.: On quadratic functionals of the Brownian sheet and related processes. Stoch. Proc. Appl. 116, 493–538 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  7. Hu, Y., Nualart, D.: Renormalized self-intersection local time for fractional Brownian motion. Ann. Probab. 33(3), 948–983 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  8. Itô, K.: Multiple Wiener integral. J. Math. Soc. Jpn. 3, 157–169 (1951)

    Article  MATH  Google Scholar 

  9. Jacod, J., Shiryayev, A.N.: Limit Theorems for Stochastic Processes. Springer, Berlin/Heidelberg/New York (1987)

    MATH  Google Scholar 

  10. Marinucci, D., Peccati, G.: High-frequency asymptotics for subordinated stationary fields on an Abelian compact group. Stoch. Proc. Appl. 118(4), 585–613 (2007)

    Article  MathSciNet  Google Scholar 

  11. Marinucci, D., Peccati, G.: Group representations and high-resolution central limit theorems for subordinated spherical random fields. Bernoulli. In revision (2007)

  12. Neuenkirch, A., Nourdin, I.: Exact rate of convergence of some approximation schemes associated to SDEs driven by fractional Brownian motion. J. Theoret. Probab. 20(4), 871–899 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  13. Nourdin, I.: Asymptotic behavior of weighted quadratic and cubic variations of fractional Brownian motion. Ann. Probab. 36(6), 2159–2175 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  14. Nourdin, I., Peccati, G., Reinert, G.: Invariance principles for homogeneous sums: universality of Wiener chaos. Preprint (2009)

  15. Nourdin, I., Peccati, G.: Non-central convergence of multiple integrals. Ann. Probab. 37(4), 1412–1426 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  16. Nourdin, I., Peccati, G.: Stein’s method on Wiener chaos. Probab. Theory Relat. Fields 145(1), 75–118 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  17. Nourdin, I., Peccati, G.: Weighted power variations of iterated Brownian motion. Electron. J. Probab. 13, 1229–1256 (2007)

    MathSciNet  Google Scholar 

  18. Nourdin, I., Réveillac, A.: Asymptotic behavior of weighted quadratic variations of fractional Brownian motion: the critical case H=1/4. Ann. Probab. To appear (2008)

  19. Nourdin, I., Nualart, D., Tudor, C.A.: Central and non-central limit theorems for weighted power variations of fractional Brownian motion. Ann. Inst. H. Poincaré Probab. Stat. To appear (2007)

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

  21. Nualart, D., Ortiz-Latorre, S.: Central limit theorems for multiple stochastic integrals and Malliavin calculus. Stoch. Proc. Appl. 118(4), 614–628 (2007)

    Article  MathSciNet  Google Scholar 

  22. Nualart, D., Peccati, G.: Central limit theorems for sequences of multiple stochastic integrals. Ann. Probab. 33(1), 177–193 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  23. Peccati, G.: Gaussian approximations of multiple integrals. Electron. Commun. Probab. 12, 350–364 (2007)

    MATH  MathSciNet  Google Scholar 

  24. Peccati, G., Taqqu, M.S.: Stable convergence of generalized L 2 stochastic integrals and the principle of conditioning. Electron. J. Probab. 12(15), 447–480 (2007)

    MATH  MathSciNet  Google Scholar 

  25. Peccati, G., Taqqu, M.S.: Stable convergence of multiple Wiener–Itô integrals. J. Theoret. Probab. 21(3), 527–570 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  26. Peccati, G., Taqqu, M.S.: Moments, cumulants and diagram formulae for non-linear functionals of random measures. Preprint (2008)

  27. Peccati, G., Tudor, C.A.: Gaussian limits for vector-valued multiple stochastic integrals. In: Séminaire de Probabilités XXXVIII. Lecture Notes in Math., vol. 1857, pp. 247–262. Springer, Berlin (2005)

    Google Scholar 

  28. Peccati, G., Yor, M.: Four limit theorems for quadratic functionals of Brownian motion and Brownian bridge. In: Asymptotic Methods in Stochastics. Fields Institute Communication Series, pp. 75–87. AMS, Providence (2004)

    Google Scholar 

  29. Rényi, A.: On stable sequences of events. Sankhya 25A, 293–302 (1963)

    Google Scholar 

  30. Skorokhod, A.V.: On a generalization of a stochastic integral. Theory Probab. Appl. 20, 219–233 (1975)

    Article  MATH  Google Scholar 

  31. Tudor, C.A., Viens, F.: Variations and estimators for the selfsimilarity order through Malliavin calculus. Ann. Probab. To appear (2007)

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Authors and Affiliations

  1. Laboratoire de Probabilités et Modèles Aléatoires, Université Pierre et Marie Curie, Boîte courrier 188, 4 Place Jussieu, 75252, Paris Cedex 5, France

    Ivan Nourdin

  2. Department of Mathematics, University of Kansas, Lawrence, KS, 66045, USA

    David Nualart

Authors
  1. Ivan Nourdin
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  2. David Nualart
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Correspondence to Ivan Nourdin.

Additional information

The work of D. Nualart is supported by the NSF Grant DMS-0604207.

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Nourdin, I., Nualart, D. Central Limit Theorems for Multiple Skorokhod Integrals. J Theor Probab 23, 39–64 (2010). https://doi.org/10.1007/s10959-009-0258-y

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  • DOI: https://doi.org/10.1007/s10959-009-0258-y

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