Variations and estimators for self-similarity parameters via Malliavin calculus
Ciprian A. Tudor and Frederi G. Viens
Source: Ann. Probab. Volume 37, Number 6
(2009), 2093-2134.
Abstract
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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Keywords: Multiple stochastic integral; Hermite process; fractional Brownian motion; Rosenblatt process; Malliavin calculus; noncentral limit theorem; quadratic variation; Hurst parameter; self-similarity; statistical estimation
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Permanent link to this document: http://projecteuclid.org/euclid.aop/1258380783
Digital Object Identifier: doi:10.1214/09-AOP459
Mathematical Reviews number (MathSciNet): MR2573552
Zentralblatt MATH identifier: 05708795
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