标题： 增强GARCH序列：依赖结构和渐近性 显示英文标题

标题： Augmented GARCH sequences: Dependence structure and asymptotics

摘要： 增强的GARCH模型是流行且广泛使用的ARCH过程的众多扩展的统一。 它由Duan引入，除了普通的（线性）GARCH过程外，还包含指数GARCH、幂GARCH、阈值GARCH、非对称GARCH等。 在本文中，我们研究增强的$\mathrm {GARCH}(1,1)$序列的概率结构以及在统计推断问题中出现的各种泛函的渐近分布。 我们不使用模型的马尔可夫结构和隐含的混合性质，而是利用扰动GARCH序列的独立性性质，直接将其渐近行为简化为独立随机变量的情况。 这种方法适用于非常广泛的泛函，并消除了早期理论中假设的相当严格的矩和光滑性条件。 特别是，我们推导了增强GARCH变量的幂的函数中心极限定理，推导了中心极限定理中的误差率，并在几乎最优的条件下获得了它们的经验过程的渐近结果。 显示英文摘要

摘要： The augmented GARCH model is a unification of numerous extensions of the popular and widely used ARCH process. It was introduced by Duan and besides ordinary (linear) GARCH processes, it contains exponential GARCH, power GARCH, threshold GARCH, asymmetric GARCH, etc. In this paper, we study the probabilistic structure of augmented $\mathrm {GARCH}(1,1)$ sequences and the asymptotic distribution of various functionals of the process occurring in problems of statistical inference. Instead of using the Markov structure of the model and implied mixing properties, we utilize independence properties of perturbed GARCH sequences to directly reduce their asymptotic behavior to the case of independent random variables. This method applies for a very large class of functionals and eliminates the fairly restrictive moment and smoothness conditions assumed in the earlier theory. In particular, we derive functional CLTs for powers of the augmented GARCH variables, derive the error rate in the CLT and obtain asymptotic results for their empirical processes under nearly optimal conditions.