The dependence structure for PARMA models with alpha-stable innovations

TL;DR

This paper explores the dependence structure of PARMA models with alpha-stable innovations, focusing on measures like codifference and covariation, which are periodic and asymptotically proportional under certain conditions.

Contribution

It characterizes the dependence measures for PARMA models with alpha-stable innovations, highlighting their periodicity and proportionality properties.

Findings

Both codifference and covariation are periodic.

The measures are asymptotically proportional with coefficient alpha.

The covariance function is not defined for these models.

Abstract

In this paper we investigate the dependence structure for PARMA models (i.e. ARMA models with periodic coefficients) with symmetric alpha-stable innovations. In this case the covariance function is not defined and therefore other measures of dependence have to be used. After obtaining the form of the bounded solution of the PARMA system with symmetric alpha-stable innovations, we study the codifference and the covariation -- the most popular measures of dependence defined for symmetric stable time series. We show that both considered measures are periodic. Moreover we determine the cases when the codifference and the covariation are asymptotically proportional with the coefficient of proportionality equal to alpha.