Estimating integrals of stochastic processes using space-time data

Abstract

Consider a space–time stochastic process $Z_ t(x)=S(x) + \xi_t(x)$ where $S(x)$ is a signal process defined on $\mathbb{R}^d$ and $\xi_t(x)$ represents measurement errors at time $t$. For a known measurable function $v(x)$ on $\mathbb{R}^d$ and a fixed cube $ D\subset \mathbb{R}^d$, this paper proposes a linear estimator for the stochastic integral $\int_D v(x)S(x)dx$ based on space–time observations $\{Z_t(x_i); i = 1,\ldots,n; t=1,\ldots,T\}$. 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.