Discrete‐time models with periodic behavior are useful for the description of different phenomenon. The most popular time series taking into consideration the periodicity of the real data is the periodic autoregressive moving average (PARMA) model. The PARMA models were considered in the literature from a theoretical and practical point of view. Most of the considerations related to the PARMA models are based on the assumption of the Gaussian (or finite‐variance) distribution of the noise. However, in many applications, the Gaussian distribution seems to be inappropriate. Thus, generalized models are considered. The natural extension of the Gaussian distribution is the $\alpha$‐stable one which is a perfect distribution for the modeling of real data with large observations. However, for the $\alpha$‐stable‐based models the classical methods adequate to Gaussian‐based systems cannot be used. The main problem comes from the fact that, in general, for the $\alpha$‐stable based models the covariance cannot be applied as a measure of dependence. Thus, alternative measures are used. In this article, we consider the generalization of the classical PARMA models and take into consideration the $\alpha$‐stable PAR system. Moreover, we analyze the bidimensional version of the univariate model and examine its structure of cross‐dependence in the language of the alternative cross‐dependence measures appropriate for the infinite‐variance systems. As the main result, we prove that the ratio of two considered alternative cross‐dependence measures tends to the stability index of the noise distribution. This result is the continuation of the authors' previous research where a similar study was performed for one‐dimensional models based on the $\alpha$‐stable distribution. Moreover, in the authors' recent papers the stationary bidimensional time series models were considered in the same direction. Finally, we propose a possible application of the introduced methodology.

中文翻译：

---

具有α稳定分布的二维周期AR（1）模型的交叉依赖度量

具有周期性行为的离散时间模型对于描述不同现象非常有用。考虑到实际数据的周期性的最流行的时间序列是周期性自回归移动平均（PARMA）模型。从理论和实践的角度来看，文献中都考虑了PARMA模型。与PARMA模型有关的大多数考虑都是基于噪声的高斯分布（或有限方差）的假设。但是，在许多应用中，高斯分布似乎是不合适的。因此，考虑了广义模型。高斯分布的自然扩展是$\alpha$稳定的模型，它是对具有大观测值的真实数据进行建模的理想分布。但是，对于$\alpha$基于稳定的模型无法使用适合基于高斯系统的经典方法。主要问题来自以下事实：通常，对于$\alpha$基于稳定的模型，协方差不能用作依赖的度量。因此，使用了替代措施。在本文中，我们考虑了经典PARMA模型的一般化，并考虑了$\alpha$稳定的PAR系统。此外，我们分析了单变量模型的二维版本，并以适用于无限方差系统的替代交叉依赖度量的语言来检验其交叉依赖的结构。作为主要结果，我们证明了两种考虑的交叉依赖测度的比值趋向于噪声分布的稳定性指标。这一结果是作者先前研究的延续，在该研究中，对基于$\alpha$分布稳定。此外，在作者的最新论文中，固定的二维时间序列模型被认为是朝同一方向发展的。最后，我们提出了引入方法的可能应用。