Almost-sure convergence of word frequencies for every parsing

Determine whether, for p-almost every point x in the stationary process, every parsing of x according to the prefix-free collection V has an asymptotic frequency for each word in V.

Background

For a finite prefix-free collection V, the paper constructs at least one parsing of almost every outcome for which each word has an asymptotic frequency. These frequencies are represented by the quantities n_a associated with an ergodic invariant measure on the parsing space, and they need not equal the original cylinder probabilities.

The unresolved issue is stronger: whether almost every point admits the relevant asymptotic word-frequency limits for all of its parsings, rather than merely for one suitably selected parsing. Resolving this would clarify the extent to which the frequency conclusions are intrinsic to the underlying stationary sequence instead of dependent on the choice of parsing.