Bootstrapping the empirical distribution function of a spatial process

In this article, we consider a stationary α-mixing random field in IR d. Under a large-sample scheme that is a mixture of the so-called "infill" and "increasing domain" asymptotics, we establish a functional central limit theorem for the empirical processes of this random field. Further, we apply a blockwise bootstrap to the samples. Under the condition that the side length of the block λl = 0(λβn) for some 0 < β < 1, where λ n is the growth rate in the increasing domain asymptotics, we show that the bootstrapped empirical process converges weakly to the same limiting Gaussian process almost surely. Extension to multivariate random fields and application to differentiable statistical functionals are also given. A spatial version of the Bernstein's inequality is developed, which may be of some independent interest. © Springer 2006.