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An Edgeworth expansion for functionals of Gaussian fields and its applications

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  • Kim, Yoon Tae
  • Park, Hyun Suk

Abstract

This paper is concerned with the rate of convergence in the normal approximation of the sequence {Fn}, where each Fn is a functional of an infinite-dimensional Gaussian field. We develop new and powerful techniques for computing the exact rate of convergence in distribution with respect to the Kolmogorov distance. As a tool for our works, the Edgeworth expansion of general orders, with an explicitly expressed remainder, will be obtained, and this remainder term will be controlled to find upper and lower bounds of the Kolmogorov distance in the case of an arbitrary sequence {Fn}. As applications, we provide the optimal fourth moment theorem of the sequence {Fn} in the case when {Fn} is a sequence of random variables living in a fixed Wiener chaos or a finite sum of Wiener chaoses. In the former case, our results show that the conditions given in this paper seem more natural and minimal than ones appeared in the previous works.

Suggested Citation

  • Kim, Yoon Tae & Park, Hyun Suk, 2018. "An Edgeworth expansion for functionals of Gaussian fields and its applications," Stochastic Processes and their Applications, Elsevier, vol. 128(12), pages 3967-3999.
    • Handle: RePEc:eee:spapps:v:128:y:2018:i:12:p:3967-3999
    DOI: 10.1016/j.spa.2018.01.006
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    References listed on IDEAS

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    1. 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.
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    Cited by:

    1. Yoon-Tae Kim & Hyun-Suk Park, 2023. "Bound for an Approximation of Invariant Density of Diffusions via Density Formula in Malliavin Calculus," Mathematics, MDPI, vol. 11(10), pages 1-18, May.

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