Functional limit theorem for moving average processes generated by dependent random variables

Abstract

Let {Xt,t≥1} be a moving average process defined byXt = ∞∑j=0bjξt-j , where {bj,j≥0} is a sequence of real numbers and {ξt, ∞＜ t ＜∞} is a doubly infinite sequence of strictly stationary φ- mixing random variables. Under conditions on {bj, j ≥0}which entail that {Xt, t ≥ 1} is either a long memory process or a linear process, we study asymptotics of Sn ( s ) = [ns]∑t=1 Xt (properly normalized). When {Xt, t≥1} is a long memory process, we establish a functional limit theorem. When {Xt, t≥1} is a linear process, we not only obtain the multi-dimensional weak convergence for {Xt, t≥1}, but also weaken the moment condition on {ξt, - ∞＜ t ＜∞} and the restriction on {bj,j≥0}. Finally, we give some applications of our results.