Stationary entrophy estimation via string matching

We prove an asymptotic relationship between certain longest match-lengths along a single realization of a stationary process and its entropy rate: Given a process X={X_n;n∈Z} and a realization x from X, we define A_i^N(x) as the length of the shortest substring, starting at x_i, that does not appear as a contiguous substring of (x_i-N,x_i-N+1 ,…,x_i-1). We consider stationary, ergodic processes that have a discrete (finite or infinite) alphabet and also satisfy the Doeblin condition (Kontoyiannis and Suhov, 1994)