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Some mixing properties of time series models. (English) Zbl 0564.62068
J. 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.
Reviewer: M.Hallin

MSC:
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
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[1] Akaike, H., Markovian representation of stochastic processes and its applications to the analysis of autoregressive moving average processes, Ann. inst. statist. math., 26, 363-387, (1974) · Zbl 0335.62058
[2] Berbee, H.C.P., Random walks with stationary increments and renewal theory, (1979), Mathematical Center Tract No. 112 Amsterdam · Zbl 0443.60083
[3] Bradley, R.C., Absolute regularity and functions of Markov chains, Stoch. proc. appl., 14, 67-77, (1983) · Zbl 0491.60028
[4] Chanda, K.C., Strong mixing properties of linear stochastic processes, J. appl. prob., 11, 401-408, (1974) · Zbl 0281.60033
[5] Gastwirth, J.L.; Rubin, H., The asymptotic distribution theory of the empirical c.d.f. for mixing stochastic processes, Ann. statist., 3, 809-824, (1975) · Zbl 0318.62016
[6] Gorodetskii, V.V., On the strong mixing property for linear sequences, Theory probab. appl., 22, 411-413, (1977) · Zbl 0377.60046
[7] Loève, M., Probability theory, (1963), Van Nostrand Princeton · Zbl 0108.14202
[8] Pham, T.D.; Tran, L.T., The strong mixing property of the autoregressive moving average time series model, Seminaire de statistique, 59-76, (1980), Grenoble
[9] Volkonskii, V.A.; Rozanov, Yu.A., Some limit theorems for random functions, part I, Theory probab. appl., 4, 178-197, (1959) · Zbl 0092.33502
[10] Wiener, N., The Fourier integral and certain of its applications, (1933), The University Press Cambridge
[11] Withers, C.S., Central limit theorems for dependent variables I, Z. wahrsch. verw. gebiete, 57, 509-534, (1981) · Zbl 0451.60027
[12] Yoshihara, K., Limiting behavior of U-statistics for stationary absolutely regular processes, Z. wahrsch. verw. geb., 35, 237-252, (1976) · Zbl 0314.60028
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