The Annals of Probability

Variations and estimators for self-similarity parameters via Malliavin calculus

Ciprian A. Tudor and Frederi G. Viens

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Using multiple stochastic integrals and the Malliavin calculus, we analyze the asymptotic behavior of quadratic variations for a specific non-Gaussian self-similar process, the Rosenblatt process. We apply our results to the design of strongly consistent statistical estimators for the self-similarity parameter H. Although, in the case of the Rosenblatt process, our estimator has non-Gaussian asymptotics for all H>1/2, we show the remarkable fact that the process’s data at time 1 can be used to construct a distinct, compensated estimator with Gaussian asymptotics for H∈(1/2, 2/3).

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Ann. Probab., Volume 37, Number 6 (2009), 2093-2134.

First available in Project Euclid: 16 November 2009

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Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems 60H05: Stochastic integrals
Secondary: 60G18: Self-similar processes 62F12: Asymptotic properties of estimators

Multiple stochastic integral Hermite process fractional Brownian motion Rosenblatt process Malliavin calculus noncentral limit theorem quadratic variation Hurst parameter self-similarity statistical estimation


Tudor, Ciprian A.; Viens, Frederi G. Variations and estimators for self-similarity parameters via Malliavin calculus. Ann. Probab. 37 (2009), no. 6, 2093--2134. doi:10.1214/09-AOP459.

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