PARMA (Periodic Autoregressive Moving Average) models, i.e. ARMA models with periodic coefficients, are an alternative to the conventional stationary time series as they allow for modeling many real-life phenomena in various areas (e.g. in hydrology, meteorology, economics, electrical engineering). PARMA models exhibit a periodic "rhythm" that is generally much more complicated than periodicity in the mean. In the case of PARMA models with Gaussian innovations the covariance function can be used to investigate the dependence structure of these processes - it turns out to be periodic with the same period as their mean.

The assumption of normality for the observations seems not to be reasonable in the number of applications, such as signal processing, telecommunications, finance, physics and chemistry, and heavy-tailed distributions seem to be more appropriate. An important class of distributions in this context is the class of stable distributions as it is a flexible class for data modeling and includes the normal distribution as a special case.

In this paper we consider PARMA models with symmetric α-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, we study the codifference (CD) and the covariation (CV) - the most popular measures of dependence that extend the covariance and are defined for symmetric α-stable time series. We show that, as in the case of Gaussian noise, in the case of stable innovations both CD and CV are periodic. Moreover we determine the cases when the codifference and the covariation are asymptotically proportional with the coefficient of proportionality equal to α. In order to illustrate theoretical results we give some examples.

• Legal notice:
  

  Copyright (c) Pielaszek Research, all rights reserved.
  The above materials, including auxiliary resources, are subject to Publisher's copyright and the Author(s) intellectual rights. Without limiting Author(s) rights under respective Copyright Transfer Agreement, no part of the above documents may be reproduced without the express written permission of Pielaszek Research, the Publisher. Express permission from the Author(s) is required to use the above materials for academic purposes, such as lectures or scientific presentations.
  In every case, proper references including Author(s) name(s) and URL of this webpage: https://science24.com/paper/6738 must be provided.

1. Financial data analysis by means of coupled continuous-time random walk in Rachev - Rüschendorf model
2. The autocorrelation and autocovariance functions – helpful tools in the modeling problem