Asymptotic behavior of unstable INAR(p) processes

TL;DR

This paper investigates the long-term behavior of unstable INAR(p) processes, showing they converge to a squared Bessel process, which differs from classical autoregressive models, with an application to crime data.

Contribution

It provides the first analysis of the asymptotic behavior of unstable INAR(p) processes, revealing their convergence to squared Bessel processes.

Findings

Unstable INAR(p) processes converge to squared Bessel processes.

The limit behavior differs from classical autoregressive processes.

Application to Boston armed robberies data demonstrates practical relevance.

Abstract

In this paper the asymptotic behavior of an unstable integer-valued autoregressive model of order p (INAR(p)) is described. Under a natural assumption it is proved that the sequence of appropriately scaled random step functions formed from an unstable INAR(p) process converges weakly towards a squared Bessel process. We note that this limit behavior is quite different from that of familiar unstable autoregressive processes of order p. An application for Boston armed robberies data set is presented.