The distribution of nonstationary autoregressive processes under general noise conditions

Abstract

In this paper we consider the long-run distribution of a multivariate autoregressive process of the form x, = An-,xn-, + noise, where the noise has an unknown (possibly nonstationary and nonindependent) distribution and A , is a (generally) time-varying transition matrix. It can easily be shown that the process x, need not have a known long-run distribution (in particular, central limit theorem effects do not generally hold). However, if the distribution of the noise approaches a known distribution as n gets large, we show that the distribution of x, may also approach a known distribution for large n. Such a setting might occur, for example, when transient effects associated with the early stages of a system's operation die out. We first present a general result that applies for arbitrary noise distributions and general A , . Several special cases are then presented that apply for noise distributions in the infinitely divisible class and/or for asymptotically constant coefficient A,. We illustrate the results on a problem in characterizing the asymptotic distribution of the estimation error in a Kalman filter.