Rényi Entropies and Large Deviations for the First Match Function

We define the first match function Tn : Cⁿ → {1, ... , n} where C is a finite alphabet. For two copies of x₁ⁿ ∈ Cⁿ, this function gives the minimum number of steps one has to slide one copy of x₁ⁿ 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.