Premium
THE DISTRIBUTION OF NONSTATIONARY AUTOREGRESSIVE PROCESSES UNDER GENERAL NOISE CONDITIONS
Author(s) -
Spall James C.
Publication year - 1993
Publication title -
journal of time series analysis
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.576
H-Index - 54
eISSN - 1467-9892
pISSN - 0143-9782
DOI - 10.1111/j.1467-9892.1993.tb00148.x
Subject(s) - mathematics , autoregressive model , noise (video) , distribution (mathematics) , asymptotic distribution , central limit theorem , statistical physics , statistics , mathematical analysis , computer science , physics , artificial intelligence , estimator , image (mathematics)
. In this paper we consider the long‐run distribution of a multivariate autoregressive process of the form x n = A n ‐1 x n ‐1 + noise, where the noise has an unknown (possibly nonstationary and nonindependent) distribution and A n is a (generally) time‐varying transition matrix. It can easily be shown that the process x n 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 n 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 n . Several special cases are then presented that apply for noise distributions in the infinitely divisible class and/or for asymptotically constant coefficient A n . We illustrate the results on a problem in characterizing the asymptotic distribution of the estimation error in a Kalman filter.