R\'enyi entropy and pattern matching for run-length encoded sequences

Abstract

In this note, we studied the asymptotic behaviour of the length of the longest common substring for run-length encoded sequences. When the original sequences are generated by an $\alpha$-mixing process with exponential decay (or $\psi$-mixing with polynomial decay), we proved that this length grows logarithmically with a coefficient depending on the R\'enyi entropy of the pushforward measure. For Bernoulli processes and Markov chains, this coefficient is computed explicitly.