FIXED-DOMAIN ASYMPTOTICS FOR SPATIAL PERIODOGRAMS

The periodogram for a spatial process observed on a lattice is often u sed to estimate the spectral density. The bases for such estimators ar e two asymptotic properties that periodograms commonly possess: (1) th e periodogram at a particular frequency is approximately unbiased for the spectral density, and (2) the correlation of the periodogram at di stinct frequencies is approximately zero. For spatial data, it is ofte n appropriate to use fixed-domain asymptotics in which the observation s get increasingly dense in some fixed region as their number increase s. Using fixed-domain asymptotics, this article shows that standard as ymptotic results for periodograms do not apply and that using the peri odogram of the raw data can yield highly misleading results. But by ap propriately filtering the data before computing the periodogram, it is possible to obtain results similar to the standard asymptotic results for spatial periodograms.