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The Breuer–Major theorem in total variation: Improved rates under minimal regularity

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  • Nourdin, Ivan
  • Nualart, David
  • Peccati, Giovanni

Abstract

In this paper we prove an estimate for the total variation distance, in the framework of the Breuer–Major theorem, using the Malliavin–Stein method, assuming the underlying function g to be once weakly differentiable with g and g′ having finite moments of order four with respect to the standard Gaussian density. This result is proved by a combination of Gebelein’s inequality and some novel estimates involving Malliavin operators.

Suggested Citation

  • Nourdin, Ivan & Nualart, David & Peccati, Giovanni, 2021. "The Breuer–Major theorem in total variation: Improved rates under minimal regularity," Stochastic Processes and their Applications, Elsevier, vol. 131(C), pages 1-20.
    • Handle: RePEc:eee:spapps:v:131:y:2021:i:c:p:1-20
    DOI: 10.1016/j.spa.2020.08.007
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    References listed on IDEAS

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    1. Nourdin, Ivan & Peccati, Giovanni & Podolskij, Mark, 2011. "Quantitative Breuer-Major theorems," Stochastic Processes and their Applications, Elsevier, vol. 121(4), pages 793-812, April.
    2. Nourdin, Ivan & Poly, Guillaume, 2013. "Convergence in total variation on Wiener chaos," Stochastic Processes and their Applications, Elsevier, vol. 123(2), pages 651-674.
    3. 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.
    4. Nualart,David & Nualart,Eulalia, 2018. "Introduction to Malliavin Calculus," Cambridge Books, Cambridge University Press, number 9781107039124.
    5. Nualart,David & Nualart,Eulalia, 2018. "Introduction to Malliavin Calculus," Cambridge Books, Cambridge University Press, number 9781107611986.
    6. Pipiras,Vladas & Taqqu,Murad S., 2017. "Long-Range Dependence and Self-Similarity," Cambridge Books, Cambridge University Press, number 9781107039469.
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