Central limit theorems for stationary random fields under weak dependence with application to ambit and mixed moving average fields
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We obtain central limit theorems for stationary random fields which are based on the use of a novel measure of dependence called θ-lex weak dependence. We discuss hereditary properties for θ-lex and η-weak dependence and illustrate the possible applications of the weak dependence notions to the study of the asymptotic properties of stationary random fields. Our general results are applied to mixed moving average fields (MMAF in short) and ambit fields. We show general conditions such that MMAF and ambit fields, with the volatility field being an MMAF or a p-dependent random field, are weakly dependent. For all the aforementioned models, we give a complete characterization of their weak dependence coefficients and sufficient conditions to obtain asymptotic normality of their sample moments. Finally, we give explicit computations of the weak dependence coefficients in the case of MSTOU and CARMA fields.
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