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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$\begingroup$ What's an example of a differentiable stationary Gaussian process? $\endgroup$– George LowtherCommented May 14, 2011 at 22:36
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2$\begingroup$ But, limits of Gaussians are Gaussian, so the answer must be yes. $\endgroup$– George LowtherCommented May 14, 2011 at 22:37
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$\begingroup$ I suppose an example would be a point rotating about the origin in $\mathbb{R}^2$ started with a symmetric normal distribution. $\endgroup$– George LowtherCommented May 14, 2011 at 22:42
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$\begingroup$ 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)? $\endgroup$– James HsiehCommented May 15, 2011 at 1:46
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2 Answers
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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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$\begingroup$ the link is no longer valid. From the URL, I am guessing it was a preprint version of this paper (sec 2 instead of 5??): tandfonline.com/doi/abs/10.1198/016214506000000041 Banerjee, S., & Gelfand, A. E. (2006). Bayesian Wombling: Curvilinear Gradient Assessment Under Spatial Process Models. Journal of the American Statistical Association, 101(476), 1487–1501. $\endgroup$– MemmingCommented May 26, 2021 at 14:43
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Malliavin calculus is the apropriate framework for your question Take $F \in \mathscr S$ (the space of functions such that all derivatives are of polynomial growth).
We define $$ DF = \sum_i \partial_i f ( W(h_1), \dots, W(h_n) ) h_i, $$ and this should be regarded as an $H=L^2$-valued r.v.
$D$ is well-defined. In particular for $F=W(h)=\int_0^1 h_t d B_t$ this is a consequence of the Ito-isometry.
