Wavelet estimation of the long memory parameter for Hermite polynomial of Gaussian processes

We consider stationary processes with long memory which are non-Gaussian and represented as Hermite polynomials of a Gaussian process. We focus on the corresponding wavelet coefficients and study the asymptotic behavior of the sum of their squares since this sum is often used for estimating the long-memory parameter. We show that the limit is not Gaussian but can be expressed using the non-Gaussian Rosenblatt process defined as a Wiener-Itô integral of order 2. This happens even if the original process is defined through a Hermite polynomial of order higher than 2.