  We consider two independent and stationary measures over $\chi^\mathbb{N}$,
where $\chi$ finite or countable alphabet. For each pair of $n$-strings in the
product space we define $T_n^{(2)}$ as the length of the shortest path
connecting one string to the other where the paths are generating by the
underlying dynamics of the measure. For ergodic measures with positive entropy
we prove that, for almost every pair of realizations $(x,y)$, $T^{(2)}_n/n$
concentrates in one, as $n$ diverges. Under mild extra conditions we prove a
large deviation principle. This principle is linked to a quantity that compute
the similarity between the two measures that we also introduce. We further
prove its existence and other properties. We also show that the fluctuations of
$T_n^{(2)}$ converge (only) in distribution to a non-degenerated distribution.
Several examples are provided for all results.
