Predicting integrals of random fields using observations on a lattice

Abstract

For a stationary random field Z on $\mathbb{R}^d$, this work studies the asymptotic behavior of predictors of $\int v(x)Z(x)dx$ based on observations on a lattice as the distance between neighbors in the lattice tends to 0. Under a mild condition on the spectral density of Z, an asymptotic expression for the mean-squared error of a predictor of $\int v(x)Z(x)dx$ based on observations on an infinite lattice is derived. For predicting integrals over the unit cube, a simple predictor based just on observations in the unit cube is shown to be asymptotically optimal if v is sufficiently smooth and Z is not too smooth. Modified predictors extend this result to smoother processes.