On estimating the memory for finitarily Markovian processes

Finitarily Markovian processes are those processes ${Xn}{n=-\infty}^{\infty}$ for which there is a finite $K$ ($K = K({Xn}{n=-\infty}^0$) such that the conditional distribution of $X1$ given the entire past is equal to the conditional distribution of $X1$ given only ${Xn}{n=1-K}^0$. 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 ${Xn}$ for $n \geq 0$ and asks for the least value $K$ such that the conditional distribution of $X{n+1}$ given ${Xi}{i=n-K+1}^n$ is the same as the conditional distribution of $X{n+1}$ given ${Xi}_{i=-\infty}^n$. We allow for finite or countably infinite alphabet size.