Restricted Random Walk Model as a New Testing Ground for the Applicability of q-Statistics

We present exact results obtained from Master Equations for the probability function P(y, T) of sums y = Sigma(T)(t=1) 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* proportional to 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) = vertical bar x/L vertical bar(a) + p with 0 < p << 1 is different from 1, although weak, but essential, deviation from the q-Gaussian does occur for a not equal 1. To assess the role of correlations we compare the T dependence of P(y, T) for the restricted random walker case with the equivalent dependenc! e for a sum y of uncorrelated variables x each distributed according to 1/g(x)