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.
Categories: math.PR cs.IT math.IT