Local Limits for String of Frozen Characters

The model we study belongs to a wide class of Markov processes called strings of characters. The model consists of a transient random walk on integers which write and re-write letters (characters) from some finite alphabet on its location. We apply the precise asymptotic theorems established for compound semi-Markov renewal process (CSRP) to study the asymptotics of the statistics of frozen characters, that is the characters on the integers that never will be visited again after some (increasing) time.