On Powers of Gaussian White Noise

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.