Hermite rank, power rank and the generalized Weierstrass transform

TL;DR

This paper explores the relationship between Hermite rank and power rank in Gaussian processes, demonstrating their equivalence and analyzing the stability of Hermite rank under level shifts using the generalized Weierstrass transform.

Contribution

It establishes the equivalence of Hermite and power ranks in Gaussian contexts and investigates the stability of Hermite rank beyond the first level using advanced mathematical transforms.

Findings

Hermite rank equals power rank in Gaussian cases

Hermite rank higher than one is unstable under level shifts

Generalized Weierstrass transform is used to analyze stability

Abstract

Using the theory of generalized Weierstrass transform, we show that the Hermite rank is identical to the power rank in the Gaussian case, and that an Hermite rank higher than one is unstable with respect to a level shift.