content_image/300553.jpg content_image/300551.jpg content_image/300552.jpg
ity measure on G ∗ such that for any set ψ$_{1}$, ...ψ$_{m}$ ∈ G the random vectors (( b, ψ$_{1}$ ) , ... ( b, ψ$_{m}$ )) have the same distribution as for some random functional ˜ b, (( ˜ b, ψ$_{1}$ ) ... ( ˜ b, ψ$_{m}$ )) . An example is a generalized Gaussian process b, so de- fined if for any ψ$_{1}$, ..., ψ$_{m}$ the joint distribution of ( b, ψ$_{1}$ ) , ..., ( b, ψ$_{m}$ ) is Gaus- sian. A generalized Gaussian process is uniquely determined by its mean functional, µ$_{b}$ : ( µ$_{b}$, ψ ) = E ( b, ψ ) , and the covariance bilinear functional, B$_{b}$ ( ψ$_{i}$, ψ$_{j}$ ) = E (( b, ψ$_{i}$ ) ( b, ψ$_{j}$ ) ) . 

Gelfand, Vilenkin (v.4, p. 260) give the covariance functional of the generalized derivative, W $^{′}$, of the Wiener process as 

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where the overbar represents complex conjugation; for real-valued processes it is not needed. The Fourier transform of a Gaussian random process b is also a Gaussian random process with covariance functional 

So for F t ( W $^{′}$) the covariance functional is 

The mean functional is zero for W ′ and F t ( W $^{′}$) . 

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Gelfand and Vilenkin (1964) provide definitions and results for general- 

