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Prediction and non-Gaussian autoregressive stationary sequences. (English) Zbl 0828.62084

The object of this paper is to show that the best one-step ahead predictor of a stationary autoregressive sequence \(x_t\), in the non- minimum phase non-Gaussian case, given \(x_j\), \(j \leq t\), is nonlinear if all moments of the innovations are finite and the roots of the associated polynomial equation are distinct. Then it is shown that the stationary autoregressive sequence is \(p\) th order Markovian, which implies that the best one-step ahead predictor in mean square in terms of the past is a function of the \(p\) preceding variables. In the non-minimum phase case the \(x_t\) process is noncausal and so the innovations are not independent of the past of the process.
The Markovian property is used to show the principal result on the nonlinearity of the best one-step ahead predictor of the process \(x_t\) in the non-minimum phase non-Gaussian case, given \(x_j\), \(j \leq t\), when all moments of the innovations are finite and the roots of the associated polynomial equation are distinct.

MSC:

62M20 Inference from stochastic processes and prediction
60G25 Prediction theory (aspects of stochastic processes)
60G10 Stationary stochastic processes
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
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