Hl. Koul et D. Surgailis, Asymptotics of empirical processes of long memory moving averages with infinite variance, STOCH PR AP, 91(2), 2001, pp. 309-336
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.