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Mixing properties of ARMA processes. (English) Zbl 0647.60042
Let \(\{\) Y(t)\(\}\) be a stationary process in R l. Denote by \({\mathcal A}_ 0\) and \({\mathcal A}_ k\) the \(\sigma\)-algebra generated by \(\{\) Y(t),t\(\leq 0\}\) and \(\{\) Y(t),t\(\geq k\}\), respectively. Define \[ \beta (k)=E\sup_{B\in {\mathcal A}\quad k}| P(B| {\mathcal A}_ 0)- P(B)|. \] If there exists \(\rho\in (0,1)\) such that \(\beta (k)=O(\rho\) k), the process \(\{\) Y(t)\(\}\) is called geometrically completely regular (GCR). The author proves that stationary vector ARMA processes are GCR (and hence strong mixing) if the innovations have absolutely continuous distribution with respect to Lebesgue measure.
Reviewer: J.Anděl

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)
Full Text: DOI
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