Structure of the third moment of the generalized Rosenblatt distribution

TL;DR

This paper derives an explicit formula for the third moment of the generalized Rosenblatt distribution, clarifying its structure and demonstrating that generalized Hermite processes form a broader class than Hermite processes.

Contribution

It provides a corrected and explicit third moment formula for the generalized Rosenblatt distribution, enhancing understanding of its structure and relation to Hermite processes.

Findings

Confirmed the class of generalized Hermite processes is strictly richer than Hermite processes

Derived a corrected explicit formula for the third moment

Numerical evaluation supports the broader class of generalized Hermite processes

Abstract

The Rosenblatt distribution appears as limit in non-central limit theorems. The generalized Rosenblatt distribution is obtained by allowing different power exponents in the kernel that defines the usual Rosenblatt distribution. We derive an explicit formula for its third moment, correcting the one in \citet{maejima:tudor:2012:selfsimilar} and \citet{tudor:2013:analysis}. Evaluating this formula numerically, we are able to confirm that the class of generalized Hermite processes is strictly richer than the class of Hermite processes.