{"id":357803,"date":"2017-01-25T13:56:28","date_gmt":"2017-01-25T21:56:28","guid":{"rendered":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/?post_type=msr-research-item&p=357803"},"modified":"2018-10-16T20:01:09","modified_gmt":"2018-10-17T03:01:09","slug":"invariant-finitary-codes-finite-expected-square-root-coding-length","status":"publish","type":"msr-research-item","link":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/publication\/invariant-finitary-codes-finite-expected-square-root-coding-length\/","title":{"rendered":"An Invariant Of Finitary Codes With Finite Expected Square Root Coding Length"},"content":{"rendered":"

Let p<\/span><\/span><\/span><\/span> and q<\/span><\/span><\/span><\/span> be probability vectors with the same entropy h<\/span><\/span><\/span><\/span>. Denote by B<\/span>(<\/span>p<\/span>)<\/span><\/span><\/span><\/span> the Bernoulli shift indexed by $\\Z$<\/span><\/span><\/span> with marginal distribution p<\/span><\/span><\/span><\/span>. Suppose that \u03d5<\/span><\/span><\/span><\/span> is a measure preserving homomorphism from B<\/span>(<\/span>p<\/span>)<\/span><\/span><\/span><\/span> to B<\/span>(<\/span>q<\/span>)<\/span><\/span><\/span><\/span>. We prove that if the coding length of \u03d5<\/span><\/span><\/span><\/span> has a finite 1\/2 moment, then \u03c3<\/span>2<\/span>p<\/span><\/span>=<\/span>\u03c3<\/span>2<\/span>q<\/span><\/span><\/span><\/span><\/span>, where \u03c3<\/span>2<\/span>p<\/span><\/span>=<\/span>\u2211<\/span>i<\/span><\/span>p<\/span>i<\/span><\/span>(<\/span>\u2212<\/span>log<\/span><\/span>p<\/span>i<\/span><\/span>\u2212<\/span>h<\/span>)<\/span>2<\/span><\/span><\/span><\/span><\/span> is the {\\dof informational variance} of p<\/span><\/span><\/span><\/span>. 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 \u03b8<\/span><<\/span>1<\/span><\/span><\/span><\/span>, we exhibit probability vectors p<\/span><\/span><\/span><\/span> and q<\/span><\/span><\/span><\/span> that are not permutations of each other, such that there exists a finitary isomorphism \u03a6<\/span><\/span><\/span><\/span> from B<\/span>(<\/span>p<\/span>)<\/span><\/span><\/span><\/span> to B<\/span>(<\/span>q<\/span>)<\/span><\/span><\/span><\/span>where the coding lengths of \u03a6<\/span><\/span><\/span><\/span> and of its inverse have a finite \u03b8<\/span><\/span><\/span><\/span> moment. We also present an extension to ergodic Markov chains.<\/p>\n","protected":false},"excerpt":{"rendered":"

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 \u03d5 is a measure preserving homomorphism from B(p) to B(q). We prove that if the coding length of \u03d5 has a finite 1\/2 moment, then \u03c32p=\u03c32q, where \u03c32p=\u2211ipi(\u2212logpi\u2212h)2 […]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr-author-ordering":null,"msr_publishername":"Cambridge University Press","msr_publisher_other":"","msr_booktitle":"","msr_chapter":"","msr_edition":"","msr_editors":"","msr_how_published":"","msr_isbn":"","msr_issue":"","msr_journal":"Ergodic Theory and Dynamical 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