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Say I have a probability space $(\Omega, \Sigma, P)$ and two Gaussian Processes over this space $X_1, X_2$ such that: \begin{align*} X_1: \mathbb{R} \times \Omega \rightarrow \mathbb{R} \\ X_2: \mathbb{R} \times \Omega \rightarrow \mathbb{R} \end{align*}

Under what conditions (if any) is the process $X'(\omega, x) = X_1(\omega, X_2(\omega, x))$ also a Gaussian Process? Does this require that $X_1,X_2$ are independent, or just jointly normal?

gigalord
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  • Are they assumed to be independent, or at least jointly normal? – Sangchul Lee Feb 15 '20 at 09:41
  • Good question. I will edit the question to express that I am also interested in knowing how these factors affect the answer. – gigalord Feb 15 '20 at 10:02
  • Actually, even under the independence assumption, I realized that we have the following counter-example: $$X_1(x,\omega)=x^2,\qquad X_2(x,\omega)=Z(\omega),$$ where $Z$ is a normal random variable. Then $X'=Z^2$ is not a Gaussian process. – Sangchul Lee Feb 15 '20 at 10:19
  • I suppose this is based on $X_1$ having zero variance everywhere, right? – gigalord Feb 15 '20 at 10:40
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    Not really. You can add to $X_1$ another independent gaussian process and still we get a non-gaussian process. The point is that, any non-linear transformation would destroy normality. – Sangchul Lee Feb 15 '20 at 10:44
  • Thanks for the context. If you write this as an answer I can accept it. – gigalord Feb 15 '20 at 13:36

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