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

TL;DR

This paper studies the shortest path length between two sequences generated by independent measures, showing concentration, large deviations, and fluctuation properties, with implications for measure similarity.

Contribution

It introduces a new measure of similarity between two measures based on shortest path lengths and establishes concentration, large deviation, and fluctuation results for these paths.

Findings

$T_n^{(2)}/n$ concentrates around one for ergodic measures with positive entropy

A large deviation principle for $T_n^{(2)}$ is established under mild conditions

Fluctuations of $T_n^{(2)}$ converge in distribution to a non-degenerate limit

Abstract

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.