May 2024 Asymptotic normality for a modified quadratic variation of the Hermite process
Antoine Ayache, Ciprian A. Tudor
Author Affiliations +
Bernoulli 30(2): 1154-1176 (May 2024). DOI: 10.3150/23-BEJ1627


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.


The authors are very grateful to the two anonymous referees for their valuable comments which have led to improvements of the manuscript.


Download Citation

Antoine Ayache. Ciprian A. Tudor. "Asymptotic normality for a modified quadratic variation of the Hermite process." Bernoulli 30 (2) 1154 - 1176, May 2024.


Received: 1 December 2022; Published: May 2024
First available in Project Euclid: 31 January 2024

MathSciNet: MR4699548
Digital Object Identifier: 10.3150/23-BEJ1627

Keywords: asymptotic normality , fractional Brownian motion , Hermite process , Hurst index estimation , multiple Wiener-Itô integrals , Ornstein-Uhlenbeck process , Stein-Malliavin calculus , strong consistency


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Vol.30 • No. 2 • May 2024
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