A wavelet analysis of the Rosenblatt process: chaos expansion and estimation of the self-similarity parameter

TL;DR

This paper employs chaos expansion and wavelet analysis to study the Rosenblatt process, deriving limit theorems and estimators for its self-similarity parameter, with applications demonstrated through simulations.

Contribution

It introduces a novel wavelet-based approach using chaos expansion for analyzing the Rosenblatt process and estimating its self-similarity index.

Findings

The wavelet coefficient statistic satisfies a non-central limit theorem for the Rosenblatt process.

Limit theorems enable the construction of consistent estimators for the self-similarity parameter.

Simulations illustrate the effectiveness of the proposed estimation method.

Abstract

By using chaos expansion into multiple stochastic integrals, we make a wavelet analysis of two self-similar stochastic processes: the fractional Brownian motion and the Rosenblatt process. We study the asymptotic behavior of the statistic based on the wavelet coefficients of these processes. Basically, when applied to a non-Gaussian process (such as the Rosenblatt process) this statistic satisfies a non-central limit theorem even when we increase the number of vanishing moments of the wavelet function. We apply our limit theorems to construct estimators for the self-similarity index and we illustrate our results by simulations.