Statistics & Probability Letters 54 (2001) 347–356 Asymptotics for moving average processes with dependent innovations Qiying Wang ∗ , Yan-Xia Lin, Chandra M. Gulati School of Maths and Applied Statistics, University of Wollongong, Northelds Avenue, Wollongong, NSW 2522, Australia Received July 1999; received in revised form August 2000 Abstract Let Xt be a moving average process dened by Xt = ∑ ∞ k=0  k  t -k ;t =1; 2;::: , where the innovation { k } is a centered sequence of random variables and { k } is a sequence of real numbers. Under conditions on { k } which entail that {Xt } is either a long memory process or a linear process, we study asymptotics of the partial sum process ∑ [ns] t =0 Xt . For a long memory process with innovations forming a martingale dierence sequence, the functional limit theorems of ∑ [ns] t =0 Xt (properly normalized) are derived. For a linear process, we give sucient conditions so that ∑ [ns] t =1 Xt (properly normalized) converges weakly to a standard Brownian motion if the corresponding ∑ [ns] k=1  k is true. The applications to fractional processes and other mixing innovations are also discussed. c  2001 Elsevier Science B.V. All rights reserved MSC: primary 60F15; secondary 60G09 Keywords: Functional limit theorem; Linear process; Long memory process; Fractionally integrated process; Moving average process 1. Introduction Dene a moving average process by X t = ∞  k =0  k  t -k ; t =1; 2;:::; (1) where the innovation { k } is a centered sequence of random variables and { k } is a sequence of real num- bers. In time-series analysis, this process is of great importance. Many important time-series models, such as the causal ARMA process (Brockwell and Davis, 1987, p. 89), have the type (1) with ∑ ∞ k =0 | k | ¡ ∞. Let Z t ;t =1; 2;::: denote a covariance stationary, purely non-deterministic time series with mean zero and autocovariance function  Z (k )= cov(Z t ;Z t -k ). More generally, as shown by Mcleod (1998) (also see ∗ Corresponding author. E-mail address: qiying@uow.edu.au (Q. Wang). 0167-7152/01/$-see front matter c  2001 Elsevier Science B.V. All rights reserved PII:S0167-7152(00)00195-4