Many real phenomena exhibit non‐Gaussian behavior. The non‐Gaussianity is manifested by impulsive behavior of the real data that can be found in both one‐dimensional and multi‐dimensional cases. Especially the multi‐dimensional datasets with non‐Gaussian behavior pose substantial analysis challenges to scientists and statisticians. In this article, we analyze the bidimensional vector autoregressive (VAR) model based on general bidimensional α‐stable distribution. This time series can be applied in modeling bidimensional data with impulsive behavior. We focus on the description of the spatio‐temporal dependence for analyzed bidimensional time series which in the considered case cannot be expressed in the language of the classical cross‐covariance or cross‐correlation function. We propose a new cross measure based on the alternative measure of dependence adequate for infinite variance processes, namely cross‐covariation. This article is an extension of the authors' previous work where the cross‐codifference was considered as the spatio‐temporal measure of the components of VAR model based on sub‐Gaussian distribution. In this article, we demonstrate that cross‐codifference and cross‐covariation can give different information about the relationships between components of bidimensional VAR models.

中文翻译：

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具有 α-稳定噪声的双变量 AR(1) 模型的时空相关性测度

许多真实现象表现出非高斯行为。非高斯性表现为真实数据的冲动行为，在一维和多维情况下都可以找到。特别是具有非高斯行为的多维数据集给科学家和统计学家带来了巨大的分析挑战。在本文中，我们分析了基于一般二维 α 稳定分布的二维向量自回归 (VAR) 模型。该时间序列可用于对具有冲动行为的二维数据进行建模。我们关注分析的二维时间序列的时空依赖性的描述，在所考虑的情况下，不能用经典的互协方差或互相关函数的语言来表达。我们基于适用于无限方差过程的替代性依赖度量提出了一种新的交叉度量，即交叉协变。本文是作者之前工作的扩展，其中交叉编码被视为基于亚高斯分布的 VAR 模型组件的时空度量。在本文中，我们证明了交叉共差和交叉协变可以提供有关二维 VAR 模型组件之间关系的不同信息。