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)

Citations:

Zbl 0614.60008; Zbl 0588.60056
Full Text: