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.

翻译：对于固定随机字段,我们使用称为$theta$-lex 弱依赖性的新的依赖度,获得中央限值理论。我们表明,这种依赖性概念比强的混合性更一般,也就是说,它适用于更广泛的模型类别。此外,我们讨论美元-tata$-lex和美元-eta$-weak依赖性的遗传属性,并举例说明在研究固定随机字段的无依赖性特性时,可能应用弱依赖性概念。我们的一般结果适用于混合移动平均字段(短的MMAF)和范围字段。我们显示了一般条件,即MMAF和范围域,因为波动字段是MMAF或美元依赖随机字段。对于上述所有模型,我们完整地描述其弱依赖性系数和充分条件,以获得其样本时的无依赖性正常状态。最后,我们明确计算MSTOU进程弱依赖性系数,并分析开发的药理理论在哪些条件下适用于CARMA字段。