Stable functional CLT for deterministic systems

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