Generalized Stochastic Processes: Linear Relations to White Noise and Orthogonal Representations

TL;DR

This paper establishes two linear relations between generalized stochastic processes and white noise, showing how any such process can be represented as a linear transformation of white noise and vice versa, using orthogonal expansions.

Contribution

It introduces new linear relations connecting generalized stochastic processes with white noise, including orthogonal series representations and conditions for transformation.

Findings

Any generalized stochastic process can be obtained from white noise via linear transformation.

Under certain conditions, a generalized process can be linearly transformed into white noise.

Generalized processes admit orthogonal series expansions with uncorrelated random variables.

Abstract

We present two linear relations between an arbitrary (real tempered second order) generalized stochastic process over $\mathbb{R}^{d}$ and White Noise processes over $\mathbb{R}^{d}$. The first is that any generalized stochastic process can be obtained as a linear transformation of a White Noise. The second indicates that, under dimensional compatibility conditions, a generalized stochastic process can be linearly transformed into a White Noise. The arguments rely on the regularity theorem for tempered distributions, which is used 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. The first linear relation obtained allows also to conclude that any generalized stochastic process has an orthogonal representation as a series expansion of deterministic tempered distributions weighted by uncorrelated random variables with summable variances. This representation is then used to conclude the second linear relation.