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

Abstract

This paper obtains a uniform reduction principle for the empirical process of a stationary moving average time series { X t } with long memory and independent and identically distributed innovations belonging to the domain of attraction of symmetric α-stable laws, 1< α<2. As a consequence, an appropriately standardized empirical process is shown to converge weakly in the uniform-topology to a degenerate process of the form f Z , where Z is a standard symmetric α-stable random variable and f is the marginal density of the underlying process. A similar result is obtained for a class of weighted empirical processes. We also show, for a large class of bounded functions h, that the limit law of (normalized) sums ∑ s=1 n h( X s ) is symmetric α-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 α-stable. This paper thus extends various well-known results of Dehling–Taqqu and Koul–Mukherjee from finite variance long memory models to infinite variance models of the above type.