  We present exact results obtained from Master Equations for the probability
function P(y,T) of sums $y=\sum_{t=1}^T x_t$ of the positions x_t of a discrete
random walker restricted to the set of integers between -L and L. We study the
asymptotic properties for large values of L and T. For a set of position
dependent transition probabilities the functional form of P(y,T) is with very
high precision represented by q-Gaussians when T assumes a certain value
$T^*\propto L^2$. The domain of y values for which the q-Gaussian apply
diverges with L. The fit to a q-Gaussian remains of very high quality even when
the exponent $a$ of the transition probability g(x)=|x/L|^a+p with 0<p<<1 is
different from 1, all though weak, but essential, deviation from the q-Gaussian
does occur for $a\neq1$. To assess the role of correlations we compare the T
dependence of P(y,T) for the restricted random walker case with the equivalent
dependence for a sum y of uncorrelated variables x each distributed according
to 1/g(x).
