Equivalence of Gaussian Measures and Prediction

The basic message of the results of 3.8 is that for interpolating a mean 0 weakly stationary random field based on observations on an infinite square lattice, the smaller the distance between neighboring observations in the lattice, the less the low frequency behavior of the spectrum matters. This suggests that if our goal is to interpolate our observations and we need to estimate the spectral density from these same observations, we should focus on getting the high frequency behavior of the spectral density as accurately as possible while not worrying so much about the low frequency behavior. Supposing that our observations and predictions will all take place in some bounded region R, a useful first question to ask is what can be done if we observe the process everywhere in R. Answering this question will put an upper bound on what one can hope to learn from some finite number of observations in R.