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 α-stable law, 1<α<2, with weights bj decaying as j−β, 1<β<2/α. We show that the empirical process (normalized by N1/αβ) weakly converges, as the sample size N increases, to the process cx+L++cx−L−, where L+,L− are independent totally skewed αβ-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 αβ-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/α<β<1 and 2/α<β studied in Koul and Surgailis (Stochastic Process. Appl. 91 (2001) 309) and Hsing (Ann. Probab. 27 (1999) 1579), respectively.