THE DISTRIBUTION OF NONSTATIONARY AUTOREGRESSIVE PROCESSES UNDER GENERAL NOISE CONDITIONS
James C. Spall · Journal of Time Series Analysis · 1993
Abstract. In this paper we consider the long‐run distribution of a multivariate autoregressive process of the form xn=An‐1xn‐1+ noise, where the noise has an unknown (possibly nonstationary and nonindependent) distribution and An is a (generally) time‐varying transition matrix. It can easily be shown that the process xn 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 xn 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 An. Several special cases are then presented that apply for noise distributions in the infinitely divisible class and/or for asymptotically constant coefficient An. We illustrate the results on a problem in characterizing the asymptotic distribution of the estimation error in a Kalman filter.