  We consider deterministic random walks on the real line driven by irrational
rotations, or equivalently, skew product extensions of a rotation by $\alpha$
where the skewing cocycle is a piecewise constant mean zero function with a
jump by one at a point $\beta$. When $\alpha$ is badly approximable and $\beta$
is badly approximable with respect to $\alpha$, we prove a Temporal Central
Limit theorem (in the terminology recently introduced by D.Dolgopyat and
O.Sarig), namely we show that for any fixed initial point, the occupancy random
variables, suitably rescaled, converge to a Gaussian random variable. This
result generalizes and extends a theorem by J. Beck for the special case when
$\alpha$ is quadratic irrational, $\beta$ is rational and the initial point is
the origin, recently reproved and then generalized to cover any initial point
using geometric renormalization arguments by Avila-Dolgopyat-Duryev-Sarig
(Israel J., 2015) and Dolgopyat-Sarig (J. Stat. Physics, 2016). We also use
renormalization, but in order to treat irrational values of $\beta$, instead of
geometric arguments, we use the renormalization associated to the continued
fraction algorithm and dynamical Ostrowski expansions. This yields a suitable
symbolic coding framework which allows us to reduce the main result to a CLT
for non homogeneous Markov chains.
