On the second order properties of the multidimensional periodogram for regularly spaced data

Recently there is growing interest in improving the level of knowledge of spatial and spatio-temporal processes using spectral techniques. The properties of the estimator of the spectral density, the periodogram, have been broadly studied under different asymptotic assumptions that imply a valuable loss of information about the behavior of the underlying process that is often observed on a grid of small size and with sparse data. In this scheme, neither increasing domain nor shrinking asymptotics applies. The goal of this paper is to study the properties of the multidimensional periodogram, under both cases of tapering and no tapering, and the assumption of finite dimensionality of the regular lattice where the process is observed. We present some theoretical results regarding the second order properties of the multidimensional periodogram. Furthermore, we show that, independent of the tapering procedure, periodogram values present a dependence structure which is not stationary and which particularly depends on weights which are proportional to the Bartlett kernel or the chosen taper.