On the Linear Extrapolation of a Continuous Homogeneous Random Field

An homogeneous random field is defined to be a family of random variables $x(s,t)$, $-\infty<s<\infty,-\infty<t< \infty$, such that
$$\mathbf M[x({s,t})]\equiv0,\lim_{\substack{h_1\to 0\\h_2\to0}}\mathbf M\left[x\left(s+h_1,t+h_2\right)-x(s,t)\right]^2=0$$
and that $B_x(s,t)=\mathbf M[x(s+m,t+n)x\overline{(m,n)}]$ exists and is independent of $m$ and $n$.
In the present paper, the extrapolation problem consists in determining the best linear prediction for the value of the field at any point in the upper half-plane by values at points in the entire lower half-plane. Various types of homogeneous random fields (singular fields, regular fields and fields of Markov type) are considered from the point of view of extrapolation theory. Necessary and sufficient conditions for these types of random fields are given in terms of spectral functions of the field. An expression for the mean square error of the optimum prediction is also obtained.