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There exist a large literature on the application of $q$-statistics to theout-of-equilibrium non-ergodic systems in which some degree of strongcorrelations exists. Here we study the distribution of first return times tozero, $P_R(0,t)$, 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 findthat for all values of $a\in[0,2]$ $P_R(0,t)$ 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 $L\rightarrow\infty$. This is a remarkable resultsince the exact analytical solution of the corresponding continuum modelrepresents $P_R(0,t)$ 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 MasterEquation, 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 thedistribution deviate from $q$-exponential. Further research is certainlyrequired to identify the reason for this result and also the applicability of$q$-statistics and its domain.

On the relevance of q-distribution functions: The return time distribution of restricted random walker