We introduce a fractional Langevin equation with α-stable noise and show that its solution $Y_{\kappa }\left(t\right),t\ge 0$ is the stationary α-stable Ornstein-Uhlenbeck-type process recently studied by Taqqu and Wolpert. We examine the asymptotic dependence structure of $Y_{\kappa }\left(t\right)$ via the measure of its codependence r(θ₁,θ₂,t). We prove that $Y_{\kappa }\left(t\right)$ is not a long-memory process in the sense of r(θ₁,θ₂,t). However, we find two natural continuous-time analogues of fractional ARIMA time series with long memory in the framework of the Langevin...