From the divergence between two measures to the shortest path between two observables

Abstract

We consider two independent and stationary measures over $\unicode[STIX]{x1D712}^{\mathbb{N}}$, where $\unicode[STIX]{x1D712}$ is a 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 of them to the other. Here the paths are generated by the underlying dynamic of the measures. If they are ergodic and have positive entropy we prove that, for almost every pair of realizations $(\mathbf{x},\mathbf{y})$, $T_{n}^{(2)}/n$ is concentrated in one, as $n$ diverges. Under mild extra conditions we prove a large-deviation principle. We also show that the fluctuations of $T_{n}^{(2)}$ converge (only) in distribution to a non-degenerate distribution. These results are all linked to a quantity that computes the similarity between those two measures. This is the so-called divergence between two measures, which is also introduced. Several examples are provided.