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We obtain central limit theorems for stationary random fields employing anovel measure of dependence called $\theta$-lex weak dependence. We show thatthis dependence notion is more general than strong mixing, i.e., it applies toa broader class of models. Moreover, we discuss hereditary properties for$\theta$-lex and $\eta$-weak dependence and illustrate the possibleapplications of the weak dependence notions to the study of the asymptoticproperties of stationary random fields. Our general results apply to mixedmoving average fields (MMAF in short) and ambit fields. We show generalconditions such that MMAF and ambit fields, with the volatility field being anMMAF or a $p$-dependent random field, are weakly dependent. For all the modelsmentioned above, we give a complete characterization of their weak dependencecoefficients and sufficient conditions to obtain the asymptotic normality oftheir sample moments. Finally, we give explicit computations of the weakdependence coefficients of MSTOU processes and analyze under which conditionsthe developed asymptotic theory applies to CARMA fields.

Central limit theorems for stationary random fields under weak dependence with application to ambit and mixed moving average fields