We present a new perfect simulation algorithm for stationary chains having unbounded variable length memory. This is the class of infinite memory chains for which the family of transition probabilities is represented by a probabilistic context tree. We do not assume any continuity condition: our condition is expressed in terms of the structure of the context tree. More precisely, the length of the contexts is a deterministic function of the distance to the last occurrence of some determined string of symbols. It turns out that the resulting class of chains can be seen as a natural extension of the class of chains having a renewal string. In particular, our chains exhibit a visible regeneration scheme.
"Chains with unbounded variable length memory: perfect simulation and a visible regeneration scheme." Adv. in Appl. Probab. 43 (3) 735 - 759, September 2011. https://doi.org/10.1239/aap/1316792668