On functional limits of short- and long-memory linear processes with GARCH(1,1) noises

This paper considers the short- and long-memory linear processes with GARCH (1,1) noises. The functional limit distributions of the partial sum and the sample autocovariances are derived when the tail index alpha is in (0, 2), equal to 2, and in (2, infinity), respectively. The partial sum weakly converges to a functional of alpha-stable process when alpha < 2 and converges to a functional of Brownian motion when alpha >= 2. When the process is of short-memory and alpha < 4, the autocovariances converge to functionals of alpha/2-stable processes; and if alpha >= 4, they converge to functionals of Brownian motions. In contrast, when the process is of long-memory, depending on alpha and beta (the parameter that characterizes the long-memory), the autocovariances converge to either (i) functionals of alpha/2-stable processes; (ii) Rosenblatt processes (indexed by beta, 1/2 < beta < 3/4); or (iii) fanctionals of Brownian motions. The rates of convergence in these limits depend on both the tail index alpha and whether or not the linear process is short- or long-memory. Our weak convergence is established on the space of cadlag functions on [0, 1] with either (i) the J(1) or the M-1 topology (Skorokhod, 1956); or (ii) the weaker form S topology (Jakubowski, 1997). Some statistical applications are also discussed. (C) 2014 Elsevier B.V. All rights reserved