On estimating the memory for finitarily Markovian processes

Finitarily Markovian processes are those processes ${X_n}_{n=-infty}^{infty}$ for which there is a finite $K$ ($K = K({X_n}_{n=-infty}^0$) such that the conditional distribution of $X_1$ given the entire past is equal to the conditional distribution of $X_1$ given only ${X_n}_{n=1-K}^0$. The least such value of $K$ is called the memory length. We give a rather complete analysis of the problems of universally estimating the least such value of $K$, both in the backward sense that we have just described and in the forward sense, where one observes successive values of ${X_n}$ for $n geq 0$ and asks for the least value $K$ such that the conditional distribution of $X_{n+1}$ given ${X_i}_{i=n-K+1}^n$ is the same as the conditional distribution of $X_{n+1}$ given ${X_i}_{i=-infty}^n$. We allow for finite or countably infinite alphabet size.