On generators with infinite entropy

Abstract

Many years ago B.S. Pitskel observed that the metric entropy of the shift transformation in the sample space of a stationary random process $X=\{X_n,\,n\in \mathbb Z\}$ with a countable number of states is equal to the conditional entropy $H(X_0|X_{-1},X_{-2},\dots)$ if $X$ is a stationary Markov chain (in which case the above conditional entropy is $H(X_0|X_{-1}))$, whether the entropy $H(X_0)$ is finite or not, while in general the statement is not true. In this note we present a class of processes for which Pitskel's observation holds, despite the fact that no of these processes is a Markov chain of some order.