Generalized Stochastic Processes as Linear Transformations of White Noise

We show that any (real) generalized stochastic process over $\mathbb{R}^{d}$ can be expressed as a linear transformation of a White Noise process over $\mathbb{R}^{d}$. The procedure is done by using the regularity theorem for tempered distributions to obtain a mean-square continuous stochastic process which is then expressed in a Karhunen-Lo\`eve expansion with respect to a convenient Hilbert space. This result also allows to conclude that any generalized stochastic process can be expressed as a series expansion of deterministic tempered distributions weighted by uncorrelated random variables with square-summable variances. A result specifying when a generalized stochastic process can be linearly transformed into a White Noise is also presented.