Asymptotic behavior of the cross-dependence measures for bidimensional AR(1) model with $\alpha $-stable noise

In this paper, we consider a bidimensional autoregressive model of order $1$ with $\alpha $-stable noise. Since in this case the classical measure of dependence known as the covariance function is not defined, the spatio-temporal dependence structure is described using the alternative measures, namely the codifference and the covariation functions. Here, we investigate the asymptotic relation between these two dependence measures applied to the description of the cross-dependence of the bidimensional model. We demonstrate the case when the dependence measures are asymptotically proportional with the coefficient of proportionality equal to the parameter $\alpha $. The theoretical results are supported by illustrating the asymptotic behavior of the dependence measures for two exemplary bidimensional $\alpha $-stable AR(1) systems.