An approach via the central limit theorem to randomization of Fourier’s transform

By using the central limit theorem, it is possible to consider the Fourier’s transform of a stochastic process with independent increments as a Gaussian random variable of which the mathematical expectation and the variance are respectively integrals of the mean and the variance of this process. One can then use this result to analyze the statistical properties of linear feedback systems in the frequency domain, that is to say in terms of transfer functions. The advantage of this approach is that one can deal with linear systems subject to squared white noises, in a very simple manner.