Estimating integrals of stochastic processes using space-time data

Consider a space-time stochastic process Z(t)(x) = S(x)+ xi(t)(x) where S(x) is a signal process defined on R-d and xi(t)(x) represents measurement errors at time t. For a known measurable function v(x) on R-d and a fixed cube D subset of R-d, this paper proposes a linear estimator for the stochastic integral integral(D) v(x)S(x)dx based on space-time observations {Z(t)(x(i)): i = 1,..., n; t = 1,..., T}. Under mild conditions, the asymptotic properties of the mean squared error of the estimator are derived as the spatial distance between spatial sampling locations tends to zero and as time T increases to infinity. Central limit theorems for the estimation error are also studied.