Large deviation property of waiting times for Markov and mixing processes

In this work, we study the asymptotic properties of the waiting time until the opening string in the realization of a process first appears in an independent realization of the same or a different process. We first establish that the normalized waiting time between two independent realizations of a single source obeys the large deviation property for a class of mixing processes. Using the method of Markov types, we extend the result to when both the sequences are realizations of two distinct irreducible and aperiodic Markov sources.