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

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.

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

Information

Published: 2009
First available in Project Euclid: 24 December 2009

zbMATH: 1326.60046
MathSciNet: MR2578831
Digital Object Identifier: 10.1214/09-EJS423

Subjects:
Primary: 60G18
Secondary: 60F05 , 60H05 , 62F12

Keywords: fractional Brownian motion , Malliavin calculus , multiple Wiener integral , Non-central limit theorem , Parameter estimation , Quadratic Variation , Rosenblatt process , self-similarity

Rights: Copyright © 2009 The Institute of Mathematical Statistics and the Bernoulli Society

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