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 of the positions xt 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*∝L2. 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, although weak, but essential, deviation from the q-Gaussian does occur for a≠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 dependence for a sum y of uncorrelated variables x each distributed according to 1/g(x).