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 Sα(ln(2),β,0) random variable.