Stable limits of empirical processes of moving averages with infinite variance

The paper obtains a functional limit theorem for the empirical process of a stationary moving average process Xt with i.i.d. innovations belonging to the domain of attraction of a symmetric [alpha]-stable law, 1<[alpha]<2, with weights bj decaying as j-[beta], 1<[beta]<2/[alpha]. We show that the empirical process (normalized by N1/[alpha][beta]) weakly converges, as the sample size N increases, to the process cx+L++cx-L-, where L+,L- are independent totally skewed [alpha][beta]-stable random variables, and cx+,cx- are some deterministic functions. We also show that, for any bounded function H, the weak limit of suitably normalized partial sums of H(Xs) is an [alpha][beta]-stable Lévy process with independent increments. This limiting behavior is quite different from the behavior of the corresponding empirical processes in the parameter regions 1/[alpha]<[beta]<1 and 2/[alpha]<[beta] studied in Koul and Surgailis (Stochastic Process. Appl. 91 (2001) 309) and Hsing (Ann. Probab. 27 (1999) 1579), respectively.