Abstract
We consider a modified quadratic variation of the Hermite process based on some well-chosen increments of this process. These special increments have the very useful property to be independent and identically distributed up to asymptotically negligible remainders. We prove that this modified quadratic variation satisfies a Central Limit Theorem and we derive its rate of convergence under the Wasserstein distance via Stein-Malliavin calculus. As a consequence, we construct, for the first time in the literature related to Hermite processes, a strongly consistent and asymptotically normal estimator for the Hurst parameter.
Funding Statement
The authors acknowledge partial support from the Labex CEMPI (ANR-11-LABX-007-01) and the GDR 3475 (Analyse Multifractale et Autosimilarité). A. Ayache also acknowledges partial support from the Australian Research Council’s Discovery Projects funding scheme (project number DP220101680). C. Tudor also acknowledges partial support from the projects ANR-22-CE40-0015, MATHAMSUD (22- MATH-08), ECOS SUD (C2107), Japan Science and Technology Agency CREST JPMJCR2115 and by a grant of the Ministry of Research, Innovation and Digitalization (Romania), CNCS-UEFISCDI, PN-III-P4-PCE-2021-0921, within PNCDI III.
Acknowledgments
The authors are very grateful to the two anonymous referees for their valuable comments which have led to improvements of the manuscript.
Citation
Antoine Ayache. Ciprian A. Tudor. "Asymptotic normality for a modified quadratic variation of the Hermite process." Bernoulli 30 (2) 1154 - 1176, May 2024. https://doi.org/10.3150/23-BEJ1627
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