Sensitivity of the Hermite rank

The Hermite rank appears in limit theorems involving long memory. We show that a 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. We also consider the case where the deterministic shift is replaced by centering with respect to the sample mean. The paper is a companion of Bai and Taqqu (2017) which discusses the instability of the Hermite rank in the statistical context.