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Strict stationarity of generalized autoregressive processes. (English) Zbl 0763.60015

The authors consider the multivariate stochastic difference equation \(X_{n+1}=A_{n+1}X_ n+B_{n+1}\), \(n\in\mathbb{Z}\), where \(X_ n\) and \(B_ n\) are random vectors in \(\mathbb{R}^ d\), \(d\geq 1\), \(A_ n\) is a random \(d\times d\) matrix, \(n\in\mathbb{Z}\), and \(\{(A_ n,B_ n),n\in\mathbb{Z}\}\) is a strictly stationary ergodic process. Assuming that \(A_ 1\) and \(B_ 1\) have only a logarithmic moment, a necessary and sufficient condition for existence of a strictly stationary solution independent of the future is given. Applications to ARMA processes and dynamic models with a state space representation are presented in a detailed manner.
The approach used is the same as that of P. Bougerol [Ann. Probab. 15, 40-74 (1987; Zbl 0614.60008)], where the special case \(B_ n=0\), \(n\in\mathbb{Z}\), was considered. The main result extends a result of A. Brandt [Adv. Appl. Probab. 18, No. 1, 211-220 (1986; Zbl 0588.60056)] for the one-dimensional case \(d=1\).

MSC:

60G10 Stationary stochastic processes
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
93E03 Stochastic systems in control theory (general)
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