The distribution of nonstationary autoregressive processes under general noise conditions

This paper considers the large-sample distribution of a multivariate autoregressive process of the form x_n=A_n-1 x_n-1+noise, where the noise has an unknown distribution and A_n is a (generally) time-varying transition matrix. It can be easily shown that the process x_n need not have a known large-sample 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_n may also approach a known distribution for large n. Such results have applications in, e.g., adaptive tracking, filtering, model validation, etc