Abstract
We develop a new technique for the proof of the fourth moment theorem on Wiener chaos to derive the bound in normal approximation of a random variable living a finite sum of Wiener chaos of a stationary Gaussian process with a positive correlaton. Thanks to newly developed techniques, an improved upper bound, expressed in terms of the fourth moment, will be obtained, compared with the one in Es-Sebaiy and Viens (Stoch Proc Appl 129:3018–3054, 2019). Our approach will be applied to the case where a random variable of functionals of Gaussian fields has a form of a power variation of a fractional Brownain motion and a polynomial variation of a fractional stationary Ornstein-Uhlenbeck process.
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Acknowledgements
We would like to thank two anonymous referees for their thorough reading of an earlier version of this manuscript and suggestions which considerably improved the exposition of our results.
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This research was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education (2019R111A3A01056116) and (NRF-2019R1F1A1063524).
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Kim, Y.T., Park, H.S. Fourth moment bound and stationary Gaussian processes with positive correlation. J. Korean Stat. Soc. 51, 172–197 (2022). https://doi.org/10.1007/s42952-021-00132-6
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DOI: https://doi.org/10.1007/s42952-021-00132-6