We show that alpha stable Lévy motions can be simulated by any ergodic and aperiodic probability preserving transformation. Namely we show: - for $0<α<1$ and every $α$ stable Lévy motion $\mathbb{W}$, there exists a function f whose partial sum process converges in distribution to $\mathbb{W}$. - for $1\leq α<2$ and every symmetric alpha stable Lévy motion $\mathbb{W}$, there exists a function f whose partial sum process converges in distribution to $\mathbb{W}$, - for $1< α<2$ and every $-1\leqβ\leq 1$ 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.