Invariance principles for tempered fractionally integrated processes

We discuss invariance principles for autoregressive tempered fractionally integrated moving averages in α-stable (1<α≤2) i.i.d. innovations and related tempered linear processes with vanishing tempering parameter limN→∞λ∕N=λ∗. We show that the limit of the partial sums process takes a different form in the weakly tempered (λ∗=0), strongly tempered (λ∗=∞), and moderately tempered (0<λ∗<∞) cases. These results are used to derive the limit distribution of the ordinary least squares estimate of AR(1) unit root with weakly, strongly, and moderately tempered moving average errors.