Generalized Stochastic Processes as Linear Transformations of White Noise

01/06/2021

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by   R. Carrizo Vergara, et al.

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We show that any (real) generalized stochastic process over ℝ^d can be expressed as a linear transformation of a White Noise process over 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-Loève 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.