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

Full-text: Open access

Abstract

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

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

Dates
First available in Project Euclid: 24 December 2009

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1261671303

Digital Object Identifier
doi:10.1214/09-EJS423

Mathematical Reviews number (MathSciNet)
MR2578831

Zentralblatt MATH identifier
1326.60046

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

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

Citation

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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