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 \\(\\phi\\) is a measure preserving homomorphism from \\(B(p)\\) to \\(B(q)\\). We prove that if the coding length of \\(\\phi\\) has a finite 1\/2 moment, then \\({\\sigma }_2^p={\\sigma }_2^q\\), where \\({\\sigma }_2^p={\\sum }_ip_i(-logp_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 \\(\\theta <1\\), we exhibit probability vectors \\(p\\) and \\(q\\) that are not permutations of each other, such that there exists a finitary isomorphism \\(\\Phi\\) from \\(B(p)\\) to \\(B(q)\\)where the coding lengths of \\(\\Phi\\) and of its inverse have a finite \\(\\theta\\) moment. We also present an extension to ergodic Markov chains.<\/p>\n","protected":false},"excerpt":{"rendered":"
Let and be probability vectors with the same entropy . Denote by the Bernoulli shift indexed by with marginal distribution . Suppose that is a measure preserving homomorphism from to . We prove that if the coding length of has a finite 1\/2 moment, then , where is the {\\dof informational variance} of . 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