A central limit theorem for a sequence of conditionally centered random fields

Abstract: A central limit theorem is established for a sum of random variables belonging to a sequence of random fields. The fields are assumed to have zero mean conditional on the past history and to satisfy certain conditional $\alpha$-mixing conditions in space or time. Exploiting conditional centering and the space-time structure, the limiting normal distribution is obtained for increasing spatial domain, increasing length of the sequence, or both of these. The theorem is very well suited for establishing asymptotic normality in the context of unbiased estimating function inference for a wide range of space-time processes. This is pertinent given the abundance of space-time data. Two examples demonstrate the applicability of the theorem.