Rényi Entropies and Large Deviations for the First Match Function

Abstract

We define the first match function T n : C n → {1, . . . , n} where C is a finite alphabet. For two copies of x n 1 ∈ C n , this function gives the minimum number of steps one has to slide one copy of x n 1 to get a match with the other one. For ergodic positive entropy processes, Saussol and coauthors proved the almost sure convergence of T n /n. We compute the large deviation properties of this function. We prove that this limit is related to the Rényi entropy function, which is also proved to exist. Our results hold under a condition easy to check which defines a large class of processes. We provide some examples.

On the shortest distance between orbits and the longest common substring problem

“…When the limit exists, we will denote it H 2 . The existence of the Rényi entropy has been proved for Bernoulli and Markov measures, Gibbs states of a Hölder-continuous potential, weakly ψ-mixing processes [30] and recently for ψ g -regular processes [1].…”

Exponential law for random subshifts of finite type

“…Even if the existence of the Rényi entropy is not known in general, it was computed in some particular cases: Bernoulli shift, Markov chains and Gibbs measure of a Hölder-continuous potential [17]. The existence was also proved for φ-mixing measures [24], for weakly ψ-mixing processes [17] and for ψ g -regular processes [1].…”

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

“…ν) which is finite since χ is finite and (3) is verified. Abadi and Cardeño [1] constructed several examples of processes of renewal type, with g equal zero and one with exponential or sub-exponential measure of cylinders which verifies (3). Measures µ which verify the classical ψ-mixing condition, for each g fixed, have ψ + µ,g constant and thus (3) holds.…”

“…If both limits are equal, we denote it by R. 1 In what follows, we provide some examples. They illustrate cases for the existence (or not) of R. Finally we state the main result of this section: a general condition in which the limiting rate exists.…”

“…But here, instead of the classical linear iteration of the sub-additivity property, we use a geometric iteration. To prove the existence of the limiting rate function R we use a kind of g-regular condition which is a version of the condition introduced in [1]. This condition was used to prove a large deviation principle for the shortest return function T n of a string to itself.…”