Wavelet-Type Expansion of the Rosenblatt Process

Abstract

The Rosenblatt process is an important example of self-similar stationary increments stochastic processes whose finite-dimensional distributions are non-Gaussian with all their moments finite. We show that the Rosenblatt process admits a wavelet-type expansion which is almost surely convergent uniformly on compact intervals and which can be thought as decorrelating the high frequencies. Our wavelet expansion of the Rosenblatt process is different from standard wavelet decompositions used in the wavelet literature. It nevertheless yields natural approximations to the Rosenblatt process, possesses a multiresolution-like structure and can be used for simulation of the Rosenblatt process in practice based on the usual Mallat-type pyramidal algorithm.