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Central limit theorems for multiple stochastic integrals and Malliavin calculus

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  • Nualart, D.
  • Ortiz-Latorre, S.

Abstract

We give a new characterization for the convergence in distribution to a standard normal law of a sequence of multiple stochastic integrals of a fixed order with variance one, in terms of the Malliavin derivatives of the sequence. We also give a new proof of the main theorem in [D. Nualart, G. Peccati, Central limit theorems for sequences of multiple stochastic integrals, Ann. Probab. 33 (2005) 177-193] using techniques of Malliavin calculus. Finally, we extend our result to the multidimensional case and prove a weak convergence result for a sequence of square integrable random vectors, giving an application.

Suggested Citation

  • Nualart, D. & Ortiz-Latorre, S., 2008. "Central limit theorems for multiple stochastic integrals and Malliavin calculus," Stochastic Processes and their Applications, Elsevier, vol. 118(4), pages 614-628, April.
  • Handle: RePEc:eee:spapps:v:118:y:2008:i:4:p:614-628
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    References listed on IDEAS

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    1. Breuer, Péter & Major, Péter, 1983. "Central limit theorems for non-linear functionals of Gaussian fields," Journal of Multivariate Analysis, Elsevier, vol. 13(3), pages 425-441, September.
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    Cited by:

    1. Xu, Weijun & Sun, Qi & Xiao, Weilin, 2012. "A new energy model to capture the behavior of energy price processes," Economic Modelling, Elsevier, vol. 29(5), pages 1585-1591.
    2. Ole E. Barndorff-Nielsen & José Manuel Corcuera & Mark Podolskij, 2009. "Multipower Variation for Brownian Semistationary Processes," CREATES Research Papers 2009-21, Department of Economics and Business Economics, Aarhus University.
    3. repec:eee:spapps:v:128:y:2018:i:2:p:404-425 is not listed on IDEAS
    4. Ehsan Azmoodeh & Lauri Viitasaari, 2015. "Parameter estimation based on discrete observations of fractional Ornstein–Uhlenbeck process of the second kind," Statistical Inference for Stochastic Processes, Springer, vol. 18(3), pages 205-227, October.
    5. Noreddine, Salim & Nourdin, Ivan, 2011. "On the Gaussian approximation of vector-valued multiple integrals," Journal of Multivariate Analysis, Elsevier, vol. 102(6), pages 1008-1017, July.
    6. Ole E. Barndorff-Nielsen & José Manuel Corcuera & Mark Podolskij, 2009. "Limit theorems for functionals of higher order differences of Brownian semi-stationary processes," CREATES Research Papers 2009-60, Department of Economics and Business Economics, Aarhus University.
    7. Viens, Frederi G., 2009. "Stein's lemma, Malliavin calculus, and tail bounds, with application to polymer fluctuation exponent," Stochastic Processes and their Applications, Elsevier, vol. 119(10), pages 3671-3698, October.
    8. Barndorff-Nielsen, Ole E. & Corcuera, José Manuel & Podolskij, Mark, 2009. "Power variation for Gaussian processes with stationary increments," Stochastic Processes and their Applications, Elsevier, vol. 119(6), pages 1845-1865, June.
    9. Hu, Yaozhong & Nualart, David, 2010. "Parameter estimation for fractional Ornstein-Uhlenbeck processes," Statistics & Probability Letters, Elsevier, vol. 80(11-12), pages 1030-1038, June.
    10. Nourdin, Ivan & Poly, Guillaume, 2013. "Convergence in total variation on Wiener chaos," Stochastic Processes and their Applications, Elsevier, vol. 123(2), pages 651-674.
    11. Eden, Richard & Víquez, Juan, 2015. "Nourdin–Peccati analysis on Wiener and Wiener–Poisson space for general distributions," Stochastic Processes and their Applications, Elsevier, vol. 125(1), pages 182-216.
    12. Harnett, Daniel & Nualart, David, 2012. "Weak convergence of the Stratonovich integral with respect to a class of Gaussian processes," Stochastic Processes and their Applications, Elsevier, vol. 122(10), pages 3460-3505.
    13. Bardet, J.-M. & Tudor, C.A., 2010. "A wavelet analysis of the Rosenblatt process: Chaos expansion and estimation of the self-similarity parameter," Stochastic Processes and their Applications, Elsevier, vol. 120(12), pages 2331-2362, December.

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