An invariant of finitary codes with finite expected square root coding length
Abstract
Description
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 $σ_p^2=σ_q^2$, where $σ_p^2=\sum_i p_i(-\log p_i-h)^2$ 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.
18 pages
18 pages