Interpolation of Homogeneous Random Fields on Discrete Groups

Abstract

Interpolation and extrapolation of stationary stochastic processes has been extensively studied by Kolmogorov, Wiener and Masani, Krein and many others. Rozanov [3] has formulated many of their results and some of his own in a very neat form. In this paper some of the basic concepts and theorems related to interpolation are investigated in the more general setting of homogeneous random fields on locally compact abelian groups. In the stationary case, the regularity and singularity of the process is determined by its behavior on the class of intervals $(-\infty, t\rbrack$. Here, since the group is not necessarily ordered, this class is replaced by an arbitrary family, $I$, of non-empty Borel sets of the group. Regularity and singularity are then defined in terms of the behavior of the field on the sets of $I$. Theorems 4.1 and 5.1 generalize Kolmogorov's minimality problem [1] and an interpolation problem studied by Yaglom [6] to groups. Theorem 4.1 is also seen to include the result of Wang Shou-Jen on interolation in $R_K$ [5]. The family $I_\infty$, introduced in Section 5, provides a natural generalization of the intervals $(-\infty, t\rbrack$ for certain processes.