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 σ2p=σ2q, where σ2p=∑ipi(−logpi−h)2 is the {\dof informational variance} of p. In this result, which sharpens a theorem of Parry (1979), 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.