  We obtain central limit theorems for stationary random fields employing a
novel measure of dependence called $\theta$-lex weak dependence. We show that
this dependence notion is more general than strong mixing, i.e., it applies to
a broader class of models. Moreover, we discuss hereditary properties for
$\theta$-lex and $\eta$-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 apply 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 models
mentioned above, we give a complete characterization of their weak dependence
coefficients and sufficient conditions to obtain the asymptotic normality of
their sample moments. Finally, we give explicit computations of the weak
dependence coefficients of MSTOU processes and analyze under which conditions
the developed asymptotic theory applies to CARMA fields.
