Entropy and Prefixes

Grassberger suggested an interesting entropy estimator, namely, n log n∑ⁿᵢ₌₁ Lⁿᵢ, where Lⁿᵢ is the shortest prefix of xᵢ, xᵢ₊₁,…, which is not a prefix of any other xⱼ, xⱼ₊₁,…, for j ≤ n. We show that this estimator is not consistent for the general ergodic process, although it is consistent for Markov chains. A weaker trimmed mean type result is proved for the general case, namely, given ε > 0, eventually almost surely all but an ε fraction of the Lⁿᵢ/log n will be within ε of $1/H$. A related Hausdorff dimension conjecture is shown to be false.