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/sub n/=A/sub n-1/x/sub n-1/+noise, where the noise has an unknown distribution and A/sub n/ is a (generally) time-varying transition matrix. It can be easily shown that the process x/sub 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/sub n/ may also approach a known distribution for large n. Such results have applications in, e.g., adaptive tracking, filtering, model validation, etc.<<ETX>>

[1]  C. Mallows.  Linear processes are nearly Gaussian , 1967, Journal of Applied Probability.

[2]  James C. Spall,et al.  Validation of state-space models from a single realization of non-Gaussian measurements , 1985 .

[3]  Trevor A Spedding,et al.  AUTOREGRESSIVE MOVING AVERAGE PROCESSES WITH NON‐NORMAL RESIDUALS , 1980 .

[4]  Ing Rj Ser.  Approximation Theorems of Mathematical Statistics , 1980 .

[5]  D. P. Gaver,et al.  First-order autoregressive gamma sequences and point processes , 1980, Advances in Applied Probability.

[6]  James C. Spall.  [That BLUP is a Good Thing: The Estimation of Random Effects]: Comment: The Kalman Filter and BLUP , 1991 .

[7]  D. Freedman,et al.  On the consistency of Bayes estimates , 1986 .

[8]  Debasis Sengupta,et al.  Efficient estimation of parameters for non-Gaussian autoregressive processes , 1989, IEEE Trans. Acoust. Speech Signal Process..

[9]  ON LINEAR PROCESSES WITH GIVEN MOMENTS , 1987 .

[10]  Asymptotic properties of autoregressive integrated moving average processes , 1975 .

[11]  Andrew M. Gleason.  Fundamentals Of Abstract Analysis , 1966 .

[12]  G. C. Tiao,et al.  Asymptotic properties of multivariate nonstationary processes with applications to autoregressions , 1990 .

[13]  J. Rothstein.  An Introduction to Probability Theory and Its Application . vol. 1. William Feller. Wiley, New York; Chapman and Hall, London, ed. 2, 1957. xv + 461 pp. Illus. $10.75. , 1959, Science.

[14]  J. Spall,et al.  Asymptotic distribution theory for the kalman filter state estimator , 1984 .

[15]  A NOTE ON THE DISTRIBUTIONS OF NON-LINEAR AUTOREGRESSIVE STOCHASTIC MODELS , 1981 .

[16]  Kai Lai Chung,et al.  A Course in Probability Theory , 1949 .

[17]  On ar(1) processes with exponential white noise , 1988 .

[18]  Y. Bar-Shalom,et al.  Variable Dimension Filter for Maneuvering Target Tracking , 1982, IEEE Transactions on Aerospace and Electronic Systems.

[19]  On the marginal distribution of a first order autoregressive process , 1984 .

[20]  Haviland Wright,et al.  Non-Gaussian series and series with non-zero means: Practical implications for time series analysis , 1982 .

[21]  Norio Watanabe.  NOTE ON THE KALMAN FILTER WITH ESTIMATED PARAMETERS , 1985 .