  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, $P_R(0,t)$, 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 find
that 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 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.
