### Abstract:

This thesis is the beginning of research into indicated spatial processes for analysing complex survey data. When modelling practical problems, it is widely accepted that a spatial process is generally described by a random ﬁeld. However, before survey methods can be applied, research into the asymptotic properties of estimators, particularly estimators based upon random ﬁelds, is essential. In this thesis, we introduce a new survey sampling method which has a potentially wide application, and is described by an indicated sampling method. This sampling strategy is appropriate for setting up estimators such as Horvitz-Thompson estimator and others. We also introduce some assumptions about the spatial structures of a population, developed from examining real situations. Based on this method and these assumptions, we develop central limit theorems, functional central limit theorems, and consistent estimators of variances on non-stationary dependent random ﬁelds. Since it is important to understand the asymptotics of a new complex survey method, for this indicated sampling method, we consider central limit theorems with the assumption of conditional independence properties. In some results, we assume the partial derivatives of functions deﬁning estimators are bounded. This assumption gives an opportunity to apply the Mean Value Theorem to the consideration of asymptotics. With our new insight into the factor md−1 in the assumptions of the central limit theorem, Theorem 3.3.1, by Guyon in 1995, we see that this is essentially an estimation of the number of the pairs which share the same dependencies. We therefore introduce a function h to stand for the number of pairs with the same dependencies. Then, with an additional assumption on the joint-blocks spatial structures, we prove the L2 consistency for the estimators of the variance of the population. We then generalize the results on estimating the variance by Carlstein in 1986 and Fuller’s central limit theorem, Theorem 1.3.2, in his book in 2011. It is rare to see functional central limit theorems on non-stationary dependent random ﬁelds. This is because it is hard to verify the tightness in the high dimensional random ﬁelds. By using two criteria introduced by Billingsley in 1968, one of which, Theorem 15.6, is rarely used by other scholars, in addition to the assumptions on the nested spatial structures and the proper estimation of the fourth moment of the sample sum, we provide some original results on functional central limit theorems, where the estimation of the fourth moment develops Rio’s result, Theorem 2.1, in his report in 2013.