We consider the strongly consistent question for model selection in a large class of causal time series models, including AR($\infty$), ARCH($\infty$), TARCH($\infty$), ARMA-GARCH and many classical others processes. We propose a penalized criterion based on the quasi likelihood of the model. We provide sufficient conditions that ensure the strong consistency of the proposed procedure. Also, the estimator of the parameter of the selected model obeys the law of iterated logarithm. It appears that, unlike the result of the weak consistency obtained by Bardet {\it et al.} \cite{Bardet2020}, a dependence between the regularization parameter and the model structure is not needed.

中文翻译：

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一般因果时间序列的强一致性模型选择

我们在一大类因果时间序列模型中考虑模型选择的强一致性问题，包括 AR($\infty$)、ARCH($\infty$)、TARCH($\infty$)、ARMA-GARCH 和许多经典的其他进程。我们提出了一个基于模型的准似然的惩罚标准。我们提供了充分的条件来确保所提议程序的强一致性。此外，所选模型的参数估计量遵循迭代对数定律。看起来，与 Bardet {\it et al.} \cite{Bardet2020} 获得的弱一致性的结果不同，正则化参数和模型结构之间的依赖关系是不需要的。