There exist a large literature on the application of q-statistics to the out-
of-equilibrium non-ergodic systems in which some degree of strong correlations exists. Here we study the distribution of first return times to zero, PR(0; t), of a random walk on the set of integers {0, 1, 2, ..., L} with a position dependent transition probability given by |n/L|^a. We find that for all values of a ∈ [0, 2] P_R(0, t) can be fitted by q-exponentials, but only for a = 1 is P_R(0, t) given exactly by a q-exponential in the limit L → ∞. This is a remarkable result since the exact analytical solution of the corresponding continuum model represents P_R(0, t) as a sum of Bessel functions with
a smooth dependence on a from which we are unable to identify a = 1 as of special significance. However, from the high precision numerical iteration of the discrete Master Equation, we do verify that only for a = 1 is P_R(0, t) exactly a q-exponential and that a tiny departure from this parameter value makes the distribution deviate from q-exponential. Further research is certainly required to identify the reason for this result and also the applicability of q-statistics and its domain.