Context tree estimation for not necessarily finite memory processes, via BIC and MDL

The concept of context tree, usually defined for finite memory processes, is extended to arbitrary stationary ergodic processes (with finite alphabet). These context trees are not necessarily complete, and may be of infinite depth. The familiar Bayesian information criterion (BIC) and minimum description length (MDL) principles are shown to provide strongly consistent estimators of the context tree, via optimization of a criterion for hypothetical context trees of finite depth, allowed to grow with the sample size n as o(logn). Algorithms are provided to compute these estimators in O(n) time, and to compute them on-line for all i les n in o(nlogn) time