Some mixing properties of time series models.

*(English)*Zbl 0564.62068J. L. Gastwirth and H. Rubin [Ann. Stat. 3, 809-824 (1975; Zbl 0318.62016)] introduced the so-called Gastwirth and Rubin mixing condition in a study of the large sample behaviour of estimators for dependent processes, mainly because first-order autoregressive processes do not satisfy the more popular \(\phi\)-mixing condition. The Gastwirth and Rubin mixing condition is intermediate between strong mixing (which is weaker) and \(\phi\)-mixing, and contains absolute regularity as a particular case.

The present paper brings a substantial support to the statistical relevance of the Gastwirth and Rubin condition by showing that it is satisfied by usual ARMA processes. This result is of great theoretical importance in time series analysis, since the vast literature on absolute regular processes can now be used in this area.

The present paper brings a substantial support to the statistical relevance of the Gastwirth and Rubin condition by showing that it is satisfied by usual ARMA processes. This result is of great theoretical importance in time series analysis, since the vast literature on absolute regular processes can now be used in this area.

Reviewer: M.Hallin

##### MSC:

62M10 | Time series, auto-correlation, regression, etc. in statistics (GARCH) |

##### Keywords:

mixing rates; linear processes; weakly dependent processes; Gastwirth and Rubin mixing condition; absolute regularity; ARMA processes; time series; absolute regular processes
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\textit{T. D. Pham} and \textit{L. T. Tran}, Stochastic Processes Appl. 19, 297--303 (1985; Zbl 0564.62068)

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