On the extremes of the max-INAR(1) process for time series of counts

Abstract

In this paper we investigate extremal properties connected with the so-called max-INAR process of order one based on the binomial thinning operator, and marginal distribution exhibiting regularly-varying right-tail. In particular, attention is paid to the limiting distribution of the number of exceedances of high levels and the joint limiting law of the maximum and the minimum. Furthermore, we also look at the extremal behavior of the max-INAR process of order one under the assumption that its corresponding thinning parameter is random. Finally, the periodic case is also addressed.

Keywords:

• Time series of counts
• Binomial thinning operator
• Extremes

Acknowledgements

The authors thank the referees for carefully reading the article and for their comments, which greatly improved the article.

Additional information

Funding

This work was partially funded by the Foundation for Science and Technology, FCT, through national (MEC) and European structural (FEDER) funds, in the scope of the research projects UIDB/04621/2020 (CEMAT/UL) and UIDB/00127/2020 (IEETA/UA).