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Invariance principles for semi-stationary sequence of linear processes and applications to ARMA process. (English) Zbl 0829.60019
Summary: Linear semi-stationary processes which are very close to the mixingales considered by D. L. McLeish [Z. Wahrscheinlichkeitstheorie Verw. Geb. 32, 165-178 (1975; Zbl 0288.60034) and Ann. Probab. 5, 616-621 (1977; Zbl 0367.60021)] are introduced. For these processes an invariance principle is obtained with conditions both simpler and weaker than those retained by McLeish for the mixingales. Furthermore, a particular class of sequences of the linear processes called quasi-stationary that gives a framework well-adapted for asymptotic theory of ARMA processes is also considered. For these quasi-stationary sequences, an invariance principle is also obtained and applied to ARMA processes. The results are compared to those obtained by P. C. B. Phillips and V. Solo [Ann. Stat. 20, No. 2, 971-1001 (1992; Zbl 0759.60021)] who used the martingale approximating technique introduced by M. I. Gordin [Sov. Math., Dokl. 10, 1174-1176 (1969); translation from Dokl. Akad. Nauk SSSR 188, 739-741 (1969; Zbl 0212.50005)].

MSC:
60F05 Central limit and other weak theorems
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
60F17 Functional limit theorems; invariance principles
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