An invariant of finitary codes with finite expected square root coding length

Let p and q be probability vectors with the same entropy h. Denote by B(p) the Bernoulli shift indexed by Z with marginal distribution p. Suppose that φ is a measure preserving homomorphism from B(p) to B(q). We prove that if the coding length of φ has a finite 1/2 moment, then σ2 p = σ 2 q , where σ 2 p = ∑ i pi(− log pi − h)2 is the informational variance of p. In this result, the 1/2 moment cannot be replaced by a lower moment. On the other hand, for any θ < 1, we exhibit probability vectors p and q that are not permutations of each other, such that there exists a finitary isomorphism Φ from B(p) to B(q) where the coding lengths of Φ and of its inverse have a finite θ moment. We also present an extension to ergodic Markov chains.