Predicting Integrals of Stochastic Processes Using Space-Time Data

Consider a stationary spatial process Z(x) = S(x) + (x) on I R d where S(x) is the signal process and (x) represents measurement errors. This paper studies asymptotic properties of the mean squared error for predicting the stochastic integral R D v(x)S(x) dx based on space-time observations on a xed cube D I R d. The random noise process (x) is assumed to vary with time t, and the covariance structure of the process is investigated. Under mild conditions, the asymptotic behavior of the mean squared predicting error is derived as the spatial distance between spatial sampling locations tends to zero and as time T increases to innnity. The asymptotic distribution of the prediction error is also studied.