Fourier Transform of Stationary Random Functions

Let us consider random valued measures as the Fourier transform (FT) of stationary random functions. A stationary random function on real line, fi(K,w), with continuous covariance function has a spectral representation of

Here Z(k,ui) is a random function with orthogonal increments. This means that when the parameter values satisfy the following conditions [32—34,38,39,44]:

the integral in Equation 2.44 is a Stieltjes integral [17—23]. With this definition, the Fourier transform of a stationary random function does not appear as another random function but as some derivative of a random function with orthogonal increments. The integral presentation of Equation 2.44 can be generalized for the case of a 3D random function.

The Cluster Expansion of the Centered Random Function and Its Fourier Transform

If the random function ц(т) is centered its covariance is also its two-point correlation function, but this is not true for higher moments. As was shown in References 17—23, the я-point correlation functions are not simultaneously correlated. We introduce therefore the correlation functions h(r1, r2), h(r1, r2, r3), ... , h(r1, r2, ... ,rf) through the following cluster expansions [32—34,37,45,46]:

where the summation is extended over all parameters of the set 1, 2, K, p into clusters of at least two points according to Equation 2.46. From system (2.46)— (2.49), it follows that for a centered Gaussian random function, all correlation functions except the second order one vanish.

A graphic representation in terms of Mayer (called then the Feynman) diagrams described in References 17—23, 33 were helpful to describe the matter. Thus, the correlation function h(r1, r2, ... ,rp) was represented by a set ofp points connected by p lines:

The cluster expansion is then written graphically. For example,

This definition of correlation functions ensures that they vanish if the points r1, r2, ... , rf are not inside a common sphere of radius l(see References 17—23, 33). We also need the FT of the correlation function as mentioned below:

If the random function fi(r) is stationary, this is not a function but a measure concentrated in the hyperplane k1 + k2 + — + kp = 0 [32, 33, 38]. Hence, we can write

and call the ordinary functions g(k1, k2, — , kp) or simply the correlation functions in k-space. Using these functions, we can write the cluster expansion of the moments in k-space as

The moment (^(k1)^(k2) — Mkp)) is thus not only concentrated in the hyperplane k1 + k2 + . + kp = 0, but it appears as sums of products of terms that are concentrated in a hyper plane of lower dimensions [35—37, 45, 46].

 
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