# 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(r _{1},* r

_{2}),

*h(r*r

_{1},_{2}, r

_{3}), ... ,

*h(r*r

_{1},_{2}, ...

*,r*through the following cluster expansions [32—34,37,45,46]:

_{f})

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(r_{1}, r_{2}, ... ,rp) was represented by a set of*p* 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 *r*_{1}*, r*_{2}*, ...* , *r _{f}* 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 *k*_{1}* + k*_{2}* + —* + *k**p* = 0 [32, 33, 38]. Hence, we can write

and call the ordinary functions *g(k*_{1}*,* k_{2}, — , 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 (^(k_{1})^(k_{2}) — Mkp)) is thus not only concentrated in the hyperplane *k*_{1} + *k*_{2} + ^{.} + *k** _{p}* = 0, but it appears as sums of products of terms that are concentrated in a hyper plane of lower dimensions [35—37, 45, 46].