Convergence of the Best Linear Predictor of a Weakly Stationary Random Field

Suppose that {X-m,X-n}((m,n)is an element of Z2) is a centered, weakly stationary random field with spectral density function W. Let X' denote the best linear estimate of X-0,X-0 based on those X-m,X- (n) where the pair (m, n) varies over a lexicographical halfplane of Z(2). The convergence rate of the partial sums for X' is studied, in relation to the analytical properties of W. Under the assumption that W is bounded away from zero and infinity, it is shown that the faster the convergence rate, the smoother W is. To be more precise, the partial sums of X' converge at an exponential rate if and only if W has an analytic extension to a neighborhood of the torus T-2; the partial sums of X' converge at a polynomial rate if and only if W satisfies a generalized Holder condition. The convergence rate of X' is also characterized in terms of the related function Phi, where W = vertical bar Phi vertical bar(2) on T-2, the Fourier coefficients of Phi vanish outside the halfplane, and Phi has a certain optimality property.