Invariance principles for semi-stationary sequence of linear processes and applications to ARMA process

Linear semi-stationary processes which are very close to the mixingales considered by McLeish (1975, 1977) 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 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 Phillips and Solo (1992) who used the martingale approximating technique introduced by Gordin (1969).