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Thanks for your help in advance. I'm interested in understanding the properties of derivatives of a differentiable stationary Gaussian process. Specifically, is the derivative also a Gaussian process?

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What's an example of a differentiable stationary Gaussian process? – George Lowther May 14 2011 at 22:36
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 But, limits of Gaussians are Gaussian, so the answer must be yes. – George Lowther May 14 2011 at 22:37
I suppose an example would be a point rotating about the origin in $\mathbb{R}^2$ started with a symmetric normal distribution. – George Lowther May 14 2011 at 22:42
I apologize if I mis-stated the question -- I'm still learning about this area. Specifically, consider a 1D signal f(x) generated by some stationary stochastic process for which the distribution of f(x) is Gaussian, and (say) the autocorrelation is also Gaussian. What can be said about the distribution of values of f'(x)? – James Hsieh May 15 2011 at 1:46

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The answer is yes in the sense that the gradient of the mean is a GP defined jointly with the original GP. I'm sure it's discussed elsewhere, but you can find derivations in section 5 of http://www.biostat.umn.edu/~sudiptob/ResearchPapers/BGjasa06.pdf.

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