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

The purpose of this paper is to make a wavelet analysis of self-similar stochastic processes by using the techniques of the Malliavin calculus and the chaos expansion into multiple stochastic integrals. Our examples are the fractional Brownian motion and the Rosenblatt process. We study the asymptotic behavior of the statistics based on the wavelet coefficients of these processes. We find that, in the case when driven process is the Rosenblatt process, this statistics satisfy a non-central limit theorem although a part of it converges to a Gaussian limit. We also construct estimators for the self-similarity index and we illustrate our results by numerical simulations.