THE DISTRIBUTION OF NONSTATIONARY AUTOREGRESSIVE PROCESSES UNDER GENERAL NOISE CONDITIONS

. 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.

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