  We discuss invariance principles for autoregressive tempered fractionally
integrated moving averages in $\alpha$-stable $(1< \alpha \le 2)$ i.i.d.
innovations and related tempered linear processes with vanishing tempering
parameter $\lambda \sim \lambda_*/N$. We show that the limit of the partial
sums process takes a different form in the weakly tempered ($\lambda_* = 0$),
strongly tempered ($\lambda_* = \infty$), and moderately tempered ($0<\lambda_*
< \infty$) cases. These results are used to derive the limit distribution of
the OLS estimate of AR(1) unit root with weakly, strongly, and moderately
tempered moving average errors.
