# Derivative of a differentiable stationary Gaussian process

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?

• What's an example of a differentiable stationary Gaussian process? – George Lowther May 14 '11 at 22:36
• But, limits of Gaussians are Gaussian, so the answer must be yes. – George Lowther May 14 '11 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 '11 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 '11 at 1:46

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.