Sensivity of the Hermite rank

TL;DR

This paper investigates the stability of the Hermite rank in limit theorems involving long memory, showing that higher ranks are unstable under small data perturbations and establishing their coincidence with power ranks in Gaussian cases.

Contribution

It introduces a near higher order rank analysis demonstrating the instability of Hermite rank under perturbations and proves its equivalence with power rank in Gaussian settings.

Findings

Hermite rank higher than one is unstable under small transformations.

Limit theorems are sensitive to slight data shifts, especially for higher Hermite ranks.

Hermite rank coincides with power rank in Gaussian contexts.

Abstract

The Hermite rank appears in limit theorems involving long memory. We show that an Hermite rank higher than one is unstable when the data is slightly perturbed by transformations such as shift and scaling. We carry out a "near higher order rank analysis" to illustrate how the limit theorems are affected by a shift perturbation that is decreasing in size. As a byproduct of our analysis, we also prove the coincidence of the Hermite rank and the power rank in the Gaussian context. The paper is a technical companion of \citet{bai:taqqu:2017:instability} which discusses the instability of the Hermite rank in the statistical context. (Older title "Some properties of the Hermite rank">)