The time series with periodic behavior, such as the periodic autoregressive (PAR) models belonging to the class of the periodically correlated processes, are present in various real applications. In the literature, such processes were considered in different directions, especially with the Gaussian-distributed noise. However, in most of the applications, the assumption of the finite-variance distribution seems to be too simplified. Thus, one can consider the extensions of the classical PAR model where the non-Gaussian distribution is applied. In particular, the Gaussian distribution can be replaced by the infinite-variance distribution, e.g. by the $\alpha-$stable distribution. In this paper, we focus on the multidimensional $\alpha-$stable PAR time series models. For such models, we propose a new estimation method based on the Yule-Walker equations. However, since for the infinite-variance case the covariance does not exist, thus it is replaced by another measure, namely the covariation. In this paper we propose to apply two estimators of the covariation measure. The first one is based on moment representation (moment-based) while the second one - on the spectral measure representation (spectral-based). The validity of the new approaches are verified using the Monte Carlo simulations in different contexts, including the sample size and the index of stability of the noise. Moreover, we compare the moment-based covariation-based method with spectral-based covariation-based technique. Finally, the real data analysis is presented.

翻译：具有周期行为的时间序列，例如属于周期相关过程类别的周期自回归（PAR）模型，存在于各种实际应用中。在文献中，此类过程从不同方向被研究，尤其是当噪声服从高斯分布时。然而，在大多数应用中，有限方差分布的假设似乎过于简化。因此，可以考虑经典PAR模型的扩展形式，其中应用了非高斯分布。特别地，高斯分布可以被无穷方差分布替代，例如α-稳定分布。本文重点研究多维α-稳定PAR时间序列模型。针对此类模型，我们提出了一种基于Yule-Walker方程的新估计方法。然而，由于无穷方差情况下协方差不存在，因此用另一种度量——即协变差——替代。本文提出应用两种协变差度量估计量：第一种基于矩表示（基于矩），第二种基于谱测度表示（基于谱）。通过在不同情境下（包括样本大小和噪声稳定性指数）进行蒙特卡洛模拟，验证了新方法的有效性。此外，我们比较了基于矩的协变差方法与基于谱的协变差技术。最后，给出了真实数据分析。