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 αα is in (0,2)(0,2), equal to 2, and in (2,∞)(2,∞), respectively. The partial sum weakly converges to a functional of αα-stable process when α<2α<2 and converges to a functional of Brownian motion when α≥2α≥2. When the process is of short-memory and α<4α<4, the autocovariances converge to functionals of α/2α/2-stable processes; and if α≥4α≥4, they converge to functionals of Brownian motions. In contrast, when the process is of long-memory, depending on αα and ββ (the parameter that characterizes the long-memory), the autocovariances converge to either (i) functionals of α/2α/2-stable processes; (ii) Rosenblatt processes (indexed by ββ, 1/2<β<3/41/2<β<3/4); or (iii) functionals of Brownian motions. The rates of convergence in these limits depend on both the tail index αα and whether or not the linear process is short- or long-memory. Our weak convergence is established on the space of càdlàg functions on [0,1][0,1] with either (i) the J1J1 or the M1M1 topology (Skorokhod, 1956); or (ii) the weaker form SS topology (Jakubowski, 1997). Some statistical applications are also discussed.