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.

Relative Performance of Fisher Information in Interval Estimation

“…In addition, if the condition in ( 5) is also satisfied, strict inequality in (12) holds according to (11). Theorem 1 suggests that asymptotically the expected estimation error of the approximate confidence level with the expected FIM (MSE F ) never exceeds the one with the observed FIM (MSE H ), as long as the sample size n is large enough.…”

Online Stochastic Convergence Analysis of the Kalman Filter

“…This form of the estimation error is one possibility of representing the error dynamics. An autoregressive form of quantifying the estimation error is discussed in [28,29]. Another interesting possibility for estimation error quantification is presented in [6].…”

“…Therefore the time-varying parameter, , can be determined as the minimum eigenvalue of the matrix, as in (28). From the covariance prediction equation in (2), it is clear that the a priori covariance is greater than the process noise covariance matrix; therefore is always between 0 and 1.…”

“…Alternatively, increasing R will decrease . If the parameter is selected as in (28), the desired inequality (37) is satisfied, thus satisfying the homogeneous part of the problem. Next, the process noise is considered.…”

“…The value for the initial Lyapunov function upper bound, V 0 , is calculated from the assumed initial covariance matrix with (23). Additionally, the values for the time-varying convergence rate, , noise parameter, , and Lyapunov function bound are defined using (28), (29), and (24), respectively. These values are calculated online at each time step of the filter.…”