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

TL;DR

This paper investigates the asymptotic behavior of quadratic variations of the Rosenblatt process using long filters, providing exact limiting distributions and consistent estimators for its self-similarity parameter, revealing non-normal convergence.

Contribution

It introduces new formulas for the limiting distributions of quadratic variations with long filters and develops consistent estimators for the Hurst parameter of the Rosenblatt process.

Findings

Limiting distributions are explicitly derived for quadratic variations.

Estimators for the Hurst parameter are strongly consistent.

Estimators do not become asymptotically normal, unlike fractional Brownian motion.

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 \textquotedblleft non-central limit theorems\textquotedblright. 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.