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

TL;DR

This paper investigates wavelet-based methods for estimating the long memory parameter in non-Gaussian stationary processes derived from Hermite polynomials of Gaussian processes, revealing non-Gaussian limit behaviors.

Contribution

It introduces a new asymptotic analysis of wavelet coefficient sums for Hermite polynomial processes, showing convergence to the Rosenblatt process rather than Gaussian limits.

Findings

Limit distribution is Rosenblatt, not Gaussian.

Results hold for Hermite polynomials of order higher than 2.

Provides theoretical foundation for long memory estimation in non-Gaussian processes.

Abstract

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 It\^o integral of order 2. This happens even if the original process is defined through a Hermite polynomial of order higher than 2.