Stable functional CLT for deterministic systems

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