An invariant of finitary codes with finite expected square root coding length
| dc.creator | Harvey, Nate | |
| dc.creator | Peres, Yuval | |
| dc.date | 2003-09-08 | |
| dc.date.accessioned | 2026-07-07T08:17:11Z | |
| dc.date.available | 2026-07-07T08:17:11Z | |
| dc.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. | |
| dc.description | 18 pages | |
| dc.identifier | https://arxiv.org/abs/math/0309120 | |
| dc.identifier | http://arxiv.org/abs/math/0309120 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/134041 | |
| dc.subject | Probability | |
| dc.subject | Information Theory | |
| dc.title | An invariant of finitary codes with finite expected square root coding length | |
| dc.type | text |