Abstract
The Central Limit Theorem is proved for m-dependent random fields. The random field is observed in a sequence of irregular domains. The sequence of domains is increasing and at the same time the locations of the observations become more and more dense in the domains.
Highlights
In statistics most asymptotic results concern the increasing domain case
A stochastic process is observed in an increasing sequence of domains Dn where the size of Dn goes to infinity as n → ∞
In spatial statistics one can observe a random field in a fixed domain such a way that the locations of observations become dense in that domain
Summary
In statistics most asymptotic results concern the increasing domain case. That is, a stochastic process (or a random field) is observed in an increasing sequence of domains Dn where the size of Dn goes to infinity as n → ∞. When studying a continuous parameter process, we observe it at discrete points of the domain This method is called sampling (see Bosq, 1998) and it is closely related to the infill-increasing approach. The importance of the infill-increasing setup is underlined by the results of Fazekas and Chuprunov (2006) Their main theorem shows that the limiting distribution of the kernel type density estimator in the infill-increasing model can be a combination of the ones in the discrete parameter and the continuous parameter models. Park, Kim, Park, and Hwang (2009) presented central limit theorems (CLT) for irregular domains assuming nearly infill sampling. They applied the CLTs to show asymptotic normality of the kernel type density estimator. A moving average process is observed in a highly irregular two-dimensional domain and the asymptotic normality of the empirical distribution function is shown there
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