Properties of Random Fields

Abstract

This chapter provides the necessary background on random fields for understanding the subsequent chapters on prediction and inference for random fields. The focus here is on weakly stationary random fields (defined later in this section) and the associated spectral theory. Some previous exposure to Fourier methods is assumed. A knowledge of the theory of characteristic functions at the level of a graduate course in probability (see, for example, Billingsley (1995), Chung (1974), or Feller (1971)) should, for the most part, suffice. When interpolating a random field, the local behavior of the random field turn out to be critical (see Chapter 3). Accordingly, this chapter goes into considerable detail about the local behavior of random fields and its relationship to spectral theory.