弱相依平稳随机场的中心极限定理及其在边界和混合移动平均场中的应用
Central limit theorems for stationary random fields under weak dependence with application to ambit and mixed moving average fields
摘要：
利用一种称为$θ$ -lex弱依赖的新的依赖度量，得到了平稳随机场的中心极限定理.我们表明，这种依赖的概念是更普遍的比强混合，即，它适用于更广泛的模型类别。讨论了$θ$ -lex和$η$ -弱相依的遗传性质，并举例说明了弱相依概念在研究平稳随机场渐近性质中的可能应用.我们的一般结果适用于混合移动平均场（简称MMAF）和界限场。我们表明，一般条件下，MMAF和境界字段，与波动率字段是一个MMAF或$p$依赖的随机场，是弱相关的。对于上述模型，我们给出了它们的弱相关系数的一个完整刻画，以及它们的样本矩具有渐近正态性的充分条件。最后，我们给出了MSTOU过程的弱相关系数的显式计算，并分析了在何种条件下所发展的渐近理论适用于CARMA场。
英文摘要：
We obtain central limit theorems for stationary random fields employing a novel measure of dependence called $θ$-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 $θ$-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 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.
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