  We consider the problem of estimating the context tree of a stationary
ergodic process with finite alphabet without imposing additional conditions on
the process. As a starting point we introduce a Hamming metric in the space of
irreducible context trees and we use the properties of the weak topology in the
space of ergodic stationary processes to prove that if the Hamming metric is
unbounded, there exist no consistent estimators for the context tree. Even in
the bounded case we show that there exist no two-sided confidence bounds.
However we prove that one-sided inference is possible in this general setting
and we construct a consistent estimator that is a lower bound for the context
tree of the process with an explicit formula for the coverage probability. We
develop an efficient algorithm to compute the lower bound and we apply the
method to test a linguistic hypothesis about the context tree of codified
written texts in European Portuguese.
