Transient dynamics of two interacting random strings

A finite string is just a sequence of symbols from finite alphabet. We consider a Markov chain with the state space equal to the set of all pairs of strings. Transition probabilities depend only on $d$ leftmost symbols in each string. Besides that, the jumps of the chain are bounded: the lengths of strings at subsequent moments of time cannot differ by more than some $d$. We consider the case when dynamics of Markov chain is transient, i.e. as $t\to\infty$ the lengths of both strings tend to infinity with probability 1. In this situation we prove stabilization law: the distribution of symbols close to left ends of strings tends to those of some random process.