  Variable Length Memory Chains (VLMC), which are generalizations of finite
order Markov chains, turn out to be an essential tool to modelize random
sequences in many domains, as well as an interesting object in contemporary
probability theory. The question of the existence of stationary probability
measures leads us to introduce a key combinatorial structure for words produced
by a VLMC: the Longest Internal Suffix. This notion allows us to state a
necessary and sufficient condition for a general VLMC to admit a unique
invariant probability measure. This condition turns out to get a much simpler
form for a subclass of VLMC: the stable VLMC. This natural subclass, unlike the
general case, enjoys a renewal property. Namely, a stable VLMC induces a
semi-Markov chain on an at most countable state space. Unfortunately, this
discrete time renewal process does not contain the whole information of the
VLMC, preventing the study of a stable VLMC to be reduced to the study of its
induced semi-Markov chain. For a subclass of stable VLMC, the convergence in
distribution of a VLMC towards its stationary probability measure is
established. Finally, finite state space semi-Markov chains turn out to be very
special stable VLMC, shedding some new light on their limit distributions.
