Second-order Homogeneous Random Fields

(1.3) F(A) = EIZ(A)12. We assume here that the time parameter t of the process takes on all real values. For discrete parameter random processes the limits of integration in (1.1) and (1.2) must be replaced by -r to +7r. Analogous spectral representations exist for stationary processes with a multidimensional parameter t = (tl, t2, *--, tn), that is, for homogeneous random fields t(t) in an n-dimensional space Rn, and for a more general class of homogeneous fields on an arbitrary locally compact commutative group G [see formulas (2.21) to (2.23) below]. Moreover, in the case of a homogeneous field t(t) with t E Rn any additional assumptions about its symmetry impose special restrictions on the covariance function B(T) and on the spectral measures F(A) and Z(A). From the point of view of applications the most interesting is the case of a homogeneous and isotropic random field, that is, the homogeneous field t(t) which possesses spherical symmetry. The general form of the covariance function B(r), with T = JTj, of such a field in R. is given by the well-known formula of I. J. Schoenberg [1], namely