Stable functional CLT for deterministic systems

We show that -stable Lévy motions can be simulated by any ergodic and aperiodic probability-preserving transformation. Namely we show that: for 0 < α < 1 and every -stable Lévy motion, there exists a function f whose partial sum process converges in distribution to; for <1 ≤ α < 2 and every symmetric -stable Lévy motion, there exists a function f whose partial sum process converges in distribution to; for < 1< α and every there exists a function f whose associated time series is in the classical domain of attraction of an random variable.