  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">)
