On the Distribution of the Nearly Unstable AR(1) Process with Heavy Tails

Abstract

We consider a nearly unstable, or near unit root, AR(1) process with regularly varying innovations. Two different approximations for the stationary distribution of such processes exist: a Gaussian approximation arising from the nearly unstable nature of the process and a heavy-tail approximation related to the tail asymptotics of the innovations. We combine these two approximations to obtain a new uniform approximation that is valid on the entire real line. As a corollary, we obtain a precise description of the regions where each of the Gaussian and heavy-tail approximations should be used.

On random coefficient INAR(1) processes

“…It is a well-known paradigm that such an assumption yields a rich probabilistic structure of the stationary solution and allows for a great flexibility in the modeling of its asymptotic behavior. See for instance [20,22], more recent articles [9,10,24,30,37,43,44], and references therein.…”

On random coefficient INAR(1) processes

“…The assumption of heavy-tailed innovations (noise terms) in autoregressive models is quite common in the applied probability literature. See for instance [23,25], more recent articles [12,13,27,38,47,58,59], and references therein. It is a well-known paradigm that such an assumption yields a rich probabilistic structure of the stationary solution and allows for a great flexibility in the modeling of its asymptotic behavior [1,39,51,52,54].…”