An Improved Second-Order Poincaré Inequality for Functionals of Gaussian Fields

Abstract

We present an improved version of the second-order Gaussian Poincaré inequality, first introduced in Chatterjee (Probab Theory Relat Fields 143(1):1–40, 2009) and Nourdin et al. (J Funct Anal 257(2):593–609, 2009). These novel estimates are used in order to bound distributional distances between functionals of Gaussian fields and normal random variables. Several applications are developed, including quantitative central limit theorems for nonlinear functionals of stationary Gaussian fields related to the Breuer–Major theorem, improving previous findings in the literature and obtaining presumably optimal rates of convergence.

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Notes

  1. 1.

    In general, any CLT involving conditions on Hermite ranks and series of covariance coefficients is usually called a Breuer–Major Theorem, in honour of the seminal paper [2].

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Acknowledgements

This work is part of the author’s PhD thesis, defended on November 29th 2018 at the University of Luxembourg and carried out under the supervision of Giovanni Peccati. The author would like to thank him for constant help and support. The author also thanks Christian Döbler, Maurizia Rossi and Guangqu Zheng for useful discussions. Finally, the author thanks an anonymous referee that with his/her insightful and careful comments considerably improved the present paper. This work was part of the AFR research project High-dimensional challenges and non-polynomial transformations in probabilistic approximations (HIGH-NPOL) funded by FNR—Luxembourg National Research Fund.

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Correspondence to Anna Vidotto.

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Vidotto, A. An Improved Second-Order Poincaré Inequality for Functionals of Gaussian Fields. J Theor Probab 33, 396–427 (2020). https://doi.org/10.1007/s10959-019-00883-3

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Keywords

  • Central limit theorems
  • Second-order Poincaré inequalities
  • Gaussian approximation
  • Isonormal Gaussian processes
  • Functionals of Gaussian fields
  • Wigner matrices

Mathematics Subject Classification (2010)

  • 60F05
  • 60G15
  • 60H07
  • 60B20