On Powers of Gaussian White Noise

TL;DR

This paper demonstrates that powers of band-limited Gaussian processes, when properly renormalized, converge to Gaussian white noise, providing a new understanding of nonlinear transformations of white noise.

Contribution

It introduces a renormalization technique showing that powers of band-limited Gaussian processes result in Gaussian white noise, advancing the theory of nonlinear white noise transformations.

Findings

Powers of band-limited Gaussian processes, after renormalization, are Gaussian white noise.

Homogeneous polynomials of Gaussian processes remain white noise under suitable renormalization.

Provides a framework for interpreting limits of nonlinear transformations of white noise.

Abstract

Classical Gaussian white noise in communications and signal processing is viewed as the limit of zero mean second order Gaussian processes with a compactly supported flat spectral density as the support goes to infinity. The difficulty of developing a theory to deal with nonlinear transformations of white noise has been to interpret the corresponding limits. In this paper we show that a renormalization and centering of powers of band-limited Gaussian processes is Gaussian white noise and as a consequence, homogeneous polynomials under suitable renormalization remain white noises.