  We show that alpha stable L\'evy motions can be simulated by any ergodic and
aperiodic probability preserving transformation. Namely we show:
  - for $0<\alpha<1$ and every $\alpha$ stable L\'evy 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 L\'evy motion
$\mathbb{W}$, 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.
