Electronic Journal of Statistics

Variations and Hurst index estimation for a Rosenblatt process using longer filters

Alexandra Chronopoulou, Frederi G. Viens, and Ciprian A. Tudor

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The Rosenblatt process is a self-similar non-Gaussian process which lives in second Wiener chaos, and occurs as the limit of correlated random sequences in so-called “non-central limit theorems”. It shares the same covariance as fractional Brownian motion. We study the asymptotic distribution of the quadratic variations of the Rosenblatt process based on long filters, including filters based on high-order finite-difference and wavelet-based schemes. We find exact formulas for the limiting distributions, which we then use to devise strongly consistent estimators of the self-similarity parameter H. Unlike the case of fractional Brownian motion, no matter now high the filter orders are, the estimators are never asymptotically normal, converging instead in the mean square to the observed value of the Rosenblatt process at time 1.

Article information

Electron. J. Statist., Volume 3 (2009), 1393-1435.

First available in Project Euclid: 24 December 2009

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Zentralblatt MATH identifier

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

Multiple Wiener integral Rosenblatt process fractional Brownian motion non-central limit theorem quadratic variation self-similarity Malliavin calculus parameter estimation


Chronopoulou, Alexandra; Viens, Frederi G.; Tudor, Ciprian A. Variations and Hurst index estimation for a Rosenblatt process using longer filters. Electron. J. Statist. 3 (2009), 1393--1435. doi:10.1214/09-EJS423. https://projecteuclid.org/euclid.ejs/1261671303

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