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

Let ｛Xt,t≥1｝ be a moving average process defined by Xt=∑∞j=0bjξt-j, where ｛bj,j≥0｝ is a sequence of real numbers and ｛ξt,-∞＜t＜∞｝ is a doubly infinite sequence of strictly stationary ?-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=1Xt (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.

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

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.