Asymptotics of empirical processes of long memory moving averages with infinite variance

This paper obtains a uniform reduction principle for the empirical process of a stationary moving average time series {X-t} with long memory and indep endent and identically distributed innovations belonging to the domain of a ttraction of symmetric alpha -stable laws, 1 < <alpha> < 2. As a consequenc e, an appropriately standardized empirical process is shown to converge wea kly in the uniform-topology to a degenerate process of the form f Z, where Z is a standard symmetric <alpha>-stable random variable and f is the margi nal density of the underlying process. A similar result is obtained for a c lass of weighted empirical processes. We also show, for a large class of bo unded functions h, that the limit law of (normalized) sums Sigma (n)(s=1) h (X-s) is symmetric alpha -stable. An application of these results to linear regression models with moving average errors of the above type yields that a large class of M-estimators of regression parameters are asymptotically equivalent to the least-squares estimator and alpha -stable. This paper thu s extends various well-known results of Dehling-Taqqu and Koul-Mukherjee fr om finite variance long memory models to infinite variance models of the ab ove type. (C) 2001 Elsevier Science B.V. All rights reserved. MSC: primary 62G05; secondary 62J05; 62E20.